# Stabilizing multiple equilibria and cycles with noisy prediction-based
control

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Pulse stabilization of cycles with Prediction-Based Control including noise
and stochastic stabilization of maps with multiple equilibrium points is
analyzed for continuous but, generally, non-smooth maps. Sufficient conditions
of global stabilization are obtained. Introduction of noise can relax
restrictions on the control intensity. We estimate how the control can be
decreased with noise and verify it numerically.

###### Key words and phrases:

Stochastic difference equations, proportional feedback control, population
dynamics models, Beverton-Holt equation

###### 1991 Mathematics Subject Classification:

Primary: 39A50, 39A30, 93D15; Secondary: 37H10, 37H30, 93C55, 39A33.

E. Braverman is a corresponding author.The first author is supported by NSERC
grant RGPIN-2020-03934.

Elena Braverman

Dept. of Math. and Stats., University of Calgary

2500 University Drive N.W. Calgary, AB, T2N 1N4, Canada

Alexandra Rodkina

Department of Mathematics, the University of the West Indies

Mona Campus, Kingston, Jamaica

(Communicated by Martin Rasmussen)

##  1\. Introduction

Consider a difference equation

$x_{n+1}=g(x_{n}),\,\,n\in\mathbb{N},\,\,x_{0}\in\mathbb{R},$ (1.1)

with the continuous function $g:\mathbb{R}\to\mathbb{R}$ having several fixed
points in the set $\mathbb{K}:=\\{K_{j},\,j=0,1,2,\dots,j_{0}\\}$,
$K_{j_{0}}\leq\infty$. We aim to stabilize simultaneously all equilibria from
$\mathbb{K}$ with odd indexes applying Prediction-Based Control (PBC) method
[MR1732161] with variable or stochastically perturbed control
$\alpha_{n}~{}\in~{}(0,1)$

$x_{n+1}=g(x_{n})-\alpha_{n+1}(g(x_{n})-x_{n}),\quad x_{0}\in\mathbb{R},\quad
n\in{\mathbb{N}}_{0}:=\mathbb{N}\cup\\{0\\}.$ (1.2)

If $g$ is a unimodal function with a negative Schwarzian derivative,
$\alpha_{n}\equiv\alpha$ are constant, global and local stability of the
unique positive equilibrium coincide for original map (1.1), see [MR0494306],
and controlled equation (1.2), see [MR2741904]. This follows from the fact
that the controlled map inherits the same properties [MR2741904]. More
sophisticated behaviour of maps with PBC is observed if either $g$ has
multiple critical and equilibrium points, or (1.1) is considered with control
(1.2) at every $k$-th step only. This corresponds to pulse control which can
be applied in both deterministic and stochastic cases [MR4145192, MR2966855,
MR3166513]. Pulse control can be viewed as PBC for the iterated map $g^{k}$.
Even for unimodal $g$, after applying PBC, all the values at critical points
become different which does not allow to apply [MR0494306] for the pulse
control. Stabilization of equilibrium points of iterated maps with PBC
corresponds to stabilizing either an equilibrium or a cycle of the original
map.

In many practical applications, in particular for models of population
dynamics, one of two one-side Lipschitz constants for a stable equilibrium can
be less than one, while the other can be quite large. This motivates us to
concentrate on one-side constants.

For an arbitrary number $j_{0}+1$ of equilibrium points in $\mathbb{K}$, with
$g(x)-x$ changing sign at every point, starting with plus, we assume that at
each point $K_{2j+1}$, the function $g$ satisfies a one-side Lipschitz
condition:

$\begin{split}&K_{2i+1}-g(x)\leq L^{+}_{2i+1}(x-K_{2i+1}),\quad
x\in(K_{2i+1},K_{2i+2}),\\\ &g(x)-K_{2i+1}\leq L^{-}_{2i+1}(K_{2i+1}-x),\quad
x\in(K_{2i},K_{2i+1}),\end{split}\quad 2i+2\leq j_{0},$ (1.3)

where one of $L^{-}_{2i+1}$ and $L^{+}_{2i+1}$ can be infinite. The expression
$g(x)-x$ is supposed to be either non-negative or non-positive on each
$(K_{i},K_{i+1})$, and may have other equilibrium points inside
$(K_{i},K_{i+1})$. The function $g$ does not necessarily map each
$(K_{i},K_{i+1})$ and even all $(K_{0},K_{j_{0}})$ to itself.

The low threshold $L/(L+1)$ of the control is calculated, based on the minimum
$L$ of the left and the right Lipschitz constants at each $K_{2j+1}$, which
guarantees that, once a solution is in this smoother interval with a less
steep $g$, it stays there. However, this value might not be enough to prevent
a solution from overshooting, getting out of $(K_{0},K_{j_{0}})$ and from
switching between different intervals, attending some of them an infinite
number of times. Due to the sign restrictions on $g(x)-x$, a solution either
converges to one of equilibrium points or circulates infinitely between
intervals. The main goal of this paper is to find the least lower bound of the
control parameter $\alpha_{n}$ which prevents this infinite circulation.

Distinction between points $K_{2j+1}$, based on the sizes of the left or the
right Lipschitz constants, allows us to split $(K_{0},K_{j_{0}})$ into the
union of blocks of non-overlapping intervals. We show that the infinite
fluctuation of a solution is possible only inside some of those blocks, while
inside the others there is no circulation at all, after the application of the
first stage of control. For each block with possible fluctuation, we find the
least low bound for the control preventing circulation. We prove that if the
control does not exceed this bound for some block, there is a two-cycle inside
the block. The main deterministic result states that when (LABEL:cond:intr)
holds, there exists a control such that a solution $x$ to (1.2) with
$x_{0}\in(K_{0},K_{j_{0}})$, converges either to an odd-numbered equilibrium
or to an equilibrium belonging to $(K_{p},K_{p+1})\subset(K_{0},K_{j_{0}})$.

Fig. 1 illustrates the second iterate $f^{2}$ of the Ricker map
$f(x)=xe^{r(1-x)}$ with $r=2.7$. As it is seen from Fig. 1, $f^{2}$ has quite
a big Lipschitz constant on the interval $(0,K_{1})$, so the stability PBC
control parameter for it can be found based on the right-side constant.
However, since $f^{2}$ is continuously differentiable, on some interval to the
left of $K_{1}$, the Lipschitz constant is close to
$(f^{2})^{\prime}(K_{1})\approx L$. Inspired by this example, we consider a
function $g$ having four fixed points, which allows to decrease the threshold
$L/(L+1)$ of the control to the value $(L-1)/(L+1)$. We prove deterministic,
as well as stochastic version of this result.

Figure 1. The second iteration of the Ricker map for $r=2.7$.

When $\alpha_{n}$ is random, it has the form $\alpha_{n}=\alpha+\ell\xi_{n}$,
i.e. a deterministic control $\alpha$ is perturbed by a noise $\xi$ with
intensity $\ell$. We assume that random variables $\xi_{n}$, $n\in\mathbb{N}$,
are independent, identically distributed and $|\xi_{n}|\leq 1$. We also
suppose that the value of the noise $\xi_{n}$ can be however close to one with
a positive probability, which allows to apply the Borel-Cantelli lemma and
conclude that, for any however big $\mathbf{J}\in\mathbb{N}$, the value of the
noise $\xi_{n}$ can stay however close to 1 for $\mathbf{J}$ consecutive
number of steps keeping $\alpha+\ell\xi_{i}$ close to $\alpha+\ell$. This
event will happen after some random moment $\mathcal{N}$. We show that, once
$\alpha+l\xi_{n}>\underline{\beta}$, where $\underline{\beta}\in(0,1)$ is a
prescribed constant, for $n=\mathcal{N},\dots,\mathcal{N}+\mathbf{J}$ a
solution is driven into the area from where circulation between intervals
becomes impossible. This leads to convergence of a solution to one of the
equilibrium points. Thus the aim of the introduction of stochastic control is
two-fold: to demonstrate the range of noise which keeps stability for the same
interval of $\alpha_{n}$, and to improve deterministic results in a sense that
there is stability for $\ell>0$, while there is no stability in deterministic
case $\ell=0$ (compare with [MR3567271, MR3639120]). A sharp result on the
minimal value of the control is aligned with the idea of excluding a possible
two-cycle [MR0661492].

The results of the paper are novel even in a deterministic setting, for
instance, compared to the multistability case considered in [MR2966855,
Theorem 2.5]. The advantage of deterministic results in the present paper is
that one-side local Lipschitz constants are taken into account, and even in
the case of one-side infinite derivatives, stabilization with PBC can be
achieved. In addition, a solution can switch between different intervals
$(K_{p},K_{p+1})$ a finite number of times. On the other hand, in [MR2966855],
where existence of a two-side global Lipschitz constant was assumed,
stabilization was achieved when the lower bound for $\alpha$ is $1-1/L$, which
is smaller than $L/(1+L)$. However, in the case when the derivative at the
equilibrium points, where the sign $g(x)-x$ switches from positive to
negative, is finite, we extend the result from [MR2966855] and, moreover,
improve it in the stochastic case.

Note that as $\alpha_{n}$ approaches one, the controlled map in (1.2) becomes
closer to the identity map and, once one-side Lipschitz constants are finite,
stabilization is possible for $\alpha_{n}$ close enough to one, but separated
from one (otherwise, we can stabilize non-fixed points of the original map).
However, this is no longer true if one of one-side Lipschitz constants is
infinite. In addition, whenever one of these constants is quite large, which
is a typical situation for iterates of chaotic maps, choosing control
intensive enough to map each union of adjacent segments surrounding a point
that is potentially stable after a control application, can lead to
significant overshoot in estimating the required parameter. The optimal way to
find the required control bound is to trace possible two-cycles and choose a
minimal bound excluding the existence of cycles.

We present three examples which illustrate our theory, in deterministic as
well in stochastic settings. One of these result has one-side infinite
derivatives, another two are iterates of the Ricker map with $r=2.7$ and
$r=3.5$. Bifurcation diagrams for these maps demonstrate how the appropriate
noise level $\ell$ provides stability for $\alpha$ which does not work in pure
deterministic framework.

The article has the following structure. In Section 2, we consider the case of
$g$ with four equilibrium points to illustrate the main ideas, under an
assumption which combines condition (LABEL:cond:intr) and the case of
differentiable at $K_{1}$ and $K_{3}$ function $g$. In Sections 2.1-2.3 we
formulate auxiliary statements and establish both the methods and the results
for deterministic PBC with a variable control parameter. In Section 2.4 we
explore the case when the control is perturbed by an additive noise. While the
main results are stated in the text, all the proofs are deferred to the
Appendix. In Section 3 we briefly discuss the case of an arbitrary number of
equilibrium points. Most details including classification of intervals,
auxiliary statements, proofs in deterministic and stochastic cases are in the
Appendix, in particular, in Sections 6.1-6.3. Section 4 illustrates our
results with numerical examples. Discussion of the results and future research
directions in Section 5 concludes the text of the paper.

##  2\. The function with four equilibrium points

The model with four equilibrium points was inspired by the second iteration of
population dynamics models. As an example, we take a Ricker map for $r>2$ when
a two-cycle exists. Fig. 1 illustrates that, for example, the left-side
Lipschitz constant at $K_{1}$ significantly exceeds the right constant and
also the derivative at the point, leading to greater required control values.
We recall that this model corresponds to either an equilibrium or a two-cycle
stabilization with a pulse PBC control applied at every second step.

###  2.1. Auxiliary statements

######  Assumption 1.

Let $g:[K_{0},K_{4})\to[K_{0},K_{4})$, where $K_{4}\leq\infty$, be a
continuous function such that

$\begin{split}&g(K_{j})=K_{j},~{}K_{j}<K_{j+1},~{}g(x)-x>0,~{}x\in(K_{0},K_{1}),~{}g(x)>K_{0},~{}x\in(K_{0},K_{4}),\\\
&g(x)-x\mbox{ alternates its sign on adjacent
}(K_{j},K_{j+1}),\,\,j=0,1,2,3,\\\ &\mbox{ and for
some}\,\,L>1,~{}\delta\in\left[0,\min_{0\leq i\leq
3}\\{K_{i+1}-K_{i}\\}\right),\\\ &|g(x)-K_{1}|\leq
L|K_{1}-x|,~{}~{}x\in(K_{1}-\delta,K_{2}),\\\ &|K_{3}-g(x)|\leq
L|x-K_{3}|,~{}x\in(K_{2},K_{3}+\delta).\end{split}$

######  Lemma 2.1.

Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous function, $x_{n}$ be a
solution to (1.2) with some $x_{0}\in\mathbb{R}$, and $\alpha_{n}\in[0,a]$ for
some $a\in(0,1)$ and all $n\in\mathbb{N}$. If $\lim_{n\to\infty}x_{n}=x^{*}$
then $g(x^{*})=x^{*}$.

Denote

$G(v,x):=(1-v)g(x)+vx,\quad x\in\mathbb{R},\,v\in[0,1].$ (2.1)

######  Lemma 2.2.

Let $g$ satisfy Assumption 1, and $G$ be defined as in (2.1). Then,

  1. (i)

$G(1,x)=x$, $G(0,x)=g(x)$, $x\in[0,\infty)$,
$G:[0,1]\times\mathbb{R}\to\mathbb{R}$ is a continuous function;

  2. (ii)

$G(v,x)-x\to 0$ as $v\to 1$, and $G(v,x)-g(x)\to 0$ as $v\to 0$, uniformly on
$[K_{0},b]$ for each $b>K_{0}$;

  3. (iii)

for $1>a>b>0$, we have $g(x)>G(b,x)>G(a,x)>x$ if $g(x)>x$, while
$g(x)<G(b,x)<G(a,x)<x$, if $g(x)<x$;

  4. (iv)

for any $\mu_{0}\in(0,1)$, $\mu\in(\mu_{0},1)$ and
$\hat{\mu}:=\frac{\mu-\mu_{0}}{1-\mu_{0}}$, we have
$G(\mu,x)=(1-\hat{\mu})G(\mu_{0},x)+\hat{\mu}x$.

###  2.2. The first trap

We start with finding a control which forces the interval
$[K_{1}-\delta,K_{3}+\delta]$ to become invariant (a trap) in a sense that as
soon as $x_{n}\in[K_{1}-\delta,K_{3}+\delta]$ for some $n\in\mathbb{N}$, it
stays there forever and converges either to $K_{1}$ or $K_{3}$.

######  Lemma 2.3.

Let Assumption 1 hold, and $\alpha>L/(L+1)$. Let $\bar{\alpha}\in[\alpha,1)$
and $\alpha_{n}\in[\alpha,\bar{\alpha}]$ for each $n\in\mathbb{N}$. Then,

  1. (i)

$G(\alpha,\cdot):[K_{1},K_{3}]\to[K_{1},K_{3}]$;

  2. (ii)

if $\delta>0$, then, in addition to (i),
$G(\alpha,\cdot):[K_{1}-\delta,K_{1}]\to[K_{1}-\delta,K_{1}]$ and
$G(\alpha,\cdot):[K_{3},K_{3}+\delta]\to[K_{3},K_{3}+\delta]$;

  3. (iii)

for each $x_{0}\in[K_{1}-\delta,K_{2})\cup(K_{2},K_{3}+\delta]$ the solution
to (1.2) converges either to $K_{1}$ or to $K_{3}$.

Now we improve the result of Lemma 2.3 by introducing a control
$\underline{\alpha}_{0}$, which might be less than $L/(L+1)$,

$\underline{\alpha}_{0}=\left\\{\begin{array}[]{l}\frac{L}{L+1},\,\,\mbox{if}\,\,\delta=0,\\\
\inf\biggl{\\{}\beta\in\left(\frac{L-1}{L+1},1\right):\min_{x\in[K_{1}-\delta,K_{2}]}G(\beta,x)>K_{1}-\delta,\\\
\max_{x\in[K_{2},K_{3}+\delta]}G(\beta,x)<K_{3}+\delta\biggr{\\}},\,\,\mbox{if}\,\,\delta>0.\end{array}\right.$
(2.2)

######  Lemma 2.4.

Let Assumption 1 hold, $\delta>0$, $\underline{\alpha}_{0}$ be defined as in
(2.2). Then,

  1. (i)

the set introduced in (2.2) for $\delta>0$ is non-empty, so
$\underline{\alpha}_{0}$ is well defined;

  2. (ii)

$\underline{\alpha}_{0}\leq L/(L+1)$.

######  Lemma 2.5.

Let Assumption 1 hold, $\underline{\alpha}_{0}$ be defined as in (2.2),
$\alpha_{*}\in(\underline{\alpha}_{0},1)$, $\alpha^{*}\in(\alpha_{*},1)$ and
$\alpha_{n}\in[\alpha_{*},\alpha^{*}]$, for each $n\in\mathbb{N}$. Then,

  1. (i)

$G(\alpha,\cdot):(K_{1}-\delta,K_{3}+\delta)\to(K_{1}-\delta,K_{3}+\delta)$,
for each $\alpha>\underline{\alpha}_{0}$;

  2. (ii)

for $x_{0}\in(K_{1}-\delta,K_{2})$ a solution to equation (1.2) converges to
$K_{1}$, while for $x_{0}\in(K_{2},K_{3}+\delta)$ it converges to $K_{3}$.

######  Remark 1.

If $\delta=0$ and the controlled solution remains in $[K_{1},K_{3}]$, it
monotonically converges to either $K_{1}$ or $K_{3}$. Also, in Assumption 1
the function $g$ is allowed to have an infinite left-side local Lipschitz
constant (derivative) at $K_{1}$ and right-side at $K_{3}$, while the case
$\delta>0$ includes the situation when $g$ is differentiable at $K_{1}$ and
$K_{3}$.

In the case $\delta>0$ and $\alpha<L/(L+1)$, there is, generally, no
monotonicity of a solution, and to get desired stability it is not enough for
the solution to remain in $[K_{1}-\delta,K_{3}+\delta]$, there could be a
cycle inside of $(K_{1}-\delta,K_{3}+\delta)$. As expected, in this case the
control can be more flexible, and the parameter $\underline{\alpha}_{0}$ can
be chosen smaller.

The next lemma shows that sometimes the invariant set
$(K_{1}-\delta,K_{2})\cup(K_{2},K_{3}+\delta)$ can actually be extended to
$(K_{0},K_{2})\cup(K_{2},\infty)$ without increasing the low bound
$\underline{\alpha}_{0}$.

######  Lemma 2.6.

Let Assumption 1 hold, $\underline{\alpha}_{0}$ be defined as in (2.2),
$\alpha_{*}\in(\underline{\alpha}_{0},1)$, $\alpha^{*}\in(\alpha_{*},1)$ and
$\alpha_{n}\in[\alpha_{*},\alpha^{*}]$, for each $n\in\mathbb{N}$. Assume also
that for all $\alpha>\underline{\alpha}_{0}$ we have

$\sup_{x\in(K_{0},K_{1}-\delta)}G(\alpha,x)<K_{3}+\delta$ or
$\inf_{x\in(K_{3}+\delta,\infty)}G(\alpha,x)>K_{1}-\delta$. (2.3)

Then, for each $x_{0}>0$, $x_{0}\neq K_{2}$, a solution to equation (1.2)
converges to an equilibrium.

Lemma 2.6 demonstrates that if we want to keep stability of $K_{1}$ and
$K_{3}$ for all $x_{0}>0$, $x_{0}\neq K_{2}$, we need to increase the low
bound of control only if (2.3) fails. So without loss of generality, we assume
that

$g_{m}:=\max_{x\in[K_{0},K_{1}-\delta]}g(x)>K_{3}+\delta\,\,\mbox{and}\,\,g_{m}^{2}:=\min_{x\in[K_{3}+\delta,g_{m}]}g(x)<K_{1}-\delta.$
(2.4)

Condition (2.4) and Lemma 2.2 imply that, for each $\alpha\in(0,1)$,

$\max_{x\in[K_{0},K_{1}-\delta]}G(\alpha,x)\leq
g_{m}\quad\mbox{and}\,\,\min_{x\in[K_{3}+\delta,g_{m}]}G(\alpha,x)\geq
g_{m}^{2}.$

In the next section, for each $\alpha>\underline{\alpha}_{0}$, we extend the
interval $(K_{1}-\delta,K_{3}+\delta)$ keeping the property of stability of
$K_{1}$ and $K_{3}$, and then introduce the smallest $\alpha$, for which
infinite circulation of solution between $(K_{0},K_{1}-\delta)$ and
$(K_{3}+\delta,g_{m})$ becomes impossible.

###  2.3. Sequences of attracting sets

For $\delta$ as in Assumption 1 and $\beta\in[0,1)$, define now
$\underline{\kappa}(\beta)$, the largest point of maximum of $G(\beta,\cdot)$
on $(K_{0},\,K_{1}-\delta)$, and $\bar{\kappa}(\beta)$, the smallest point of
minimum of $G(\beta,\cdot)$ on $(K_{3}+\delta,\,g_{m})$:

$\begin{split}&\underline{\kappa}(\beta)=\sup\left\\{y\in\left(K_{0},\,K_{1}-\delta\right):G(\beta,y)=\sup_{x\in(K_{0},K_{1}-\delta)}G(\beta,x)\right\\},\\\
&\bar{\kappa}(\beta)=\inf\left\\{y\in\left(K_{3}+\delta,\,g_{m}\right):G(\beta,y)=\inf_{x\in(K_{3}+\delta,g_{m})}G(\beta,x)\right\\}.\end{split}$
(2.5)

Now we introduce two convergent sequences of points,
$(d_{n}(\beta))_{n\in\mathbb{N}}$ and $(c_{n}(\beta))_{n\in\mathbb{N}}$,
located in $(K_{0},K_{1}-\delta)$ and in $(K_{3}+\delta,\,g_{m})$,
respectively,

$\begin{split}&d_{0}:=K_{1}-\delta,\,c_{0}:=K_{3}+\delta,\,\,\mbox{and,
for}\,\,k\in\mathbb{N},\\\
&d_{k}(\beta):=\inf\left\\{x\in\left(K_{0},\,d_{k-1}\right):\max_{y\in[x,d_{k-1}]}G(\beta,y)\leq
c_{k-1}\right\\},\\\
&d_{k}=d_{k-1},\quad\mbox{if}\,\max_{y\in[K_{0},d_{k-1}]}G(\beta,y)\leq
c_{k-1},\\\
&c_{k}(\beta):=\sup\left\\{x\in\left(c_{k-1},\,g_{m}\right):\min_{y\in[c_{k-1},x]}G(\beta,y)\geq
d_{k}(\beta)\right\\},\\\
&c_{k}=c_{k-1},\quad\mbox{if}\quad\inf_{y\in[c_{k-1},g_{m})}G(\beta,y)\geq
d_{k}.\end{split}$ (2.6)

Assume that $k$ is the first moment for which $d_{k}=d_{k-1}$. Since
$c_{k}\geq c_{k-1}$,
$\max_{y\in[K_{0},d_{k}]}G(\beta,y)=\max_{y\in[K_{0},d_{k-1}]}G(\beta,y)\leq
c_{k-1}\leq c_{k}$, we have $d_{k+1}=d_{k}$, therefore the sequence
$(d_{n}(\beta))_{n\in\mathbb{N}}$ stops after $k-1$ steps. As
$\min_{y\in[c_{k},x]}G(\beta,y)\geq\min_{y\in[c_{k-1},x]}G(\beta,y)$ for
$x\in\left(c_{k},\,g_{m}\right)$, we get

$c_{k+1}(\beta):=\sup\left\\{x\in\left(c_{k},\,g_{m}\right):\\!\\!\\!\min_{y\in[c_{k},x]}G(\beta,y)\geq\\!\\!\\!\min_{y\in[c_{k-1},x]}G(\beta,y)\geq
d_{k}=d_{k+1}\right\\}=c_{k},$

therefore the sequence $(c_{n}(\beta))_{n\in\mathbb{N}}$ stops after $k$
steps.

Define the set of positive integers for which the iterative procedure stops

$\begin{split}&\mathcal{S}(\beta):=\\{k:d_{k}(\beta)=d_{k+1}(\beta)\,\,\mbox{or}\,\,c_{k}(\beta)=c_{k+1}(\beta)\\},\\\
&k_{0}(\beta):=\inf\\{\mathcal{S}(\beta)\\},\quad\mbox{if}\quad\mathcal{S}(\beta)\neq\emptyset,\quad
k_{0}(\beta):=\infty,\,\,\mbox{if}\,\,\mathcal{S}(\beta)=\emptyset.\end{split}$
(2.7)

The condition

$\begin{split}k_{0}(\beta):=\inf\\{k:d_{k}(\beta)=d_{k+1}(\beta)\,\,\mbox{or}\,\,c_{k}(\beta)=c_{k+1}(\beta)\\}<\infty\end{split}$
(2.8)

is essential in constructing a wider attractive set, dependent on the control
$\beta$.

The sequences $(c_{n}(\beta))_{n\in\mathbb{N}}$ and
$(d_{n}(\beta))_{n\in\mathbb{N}}$ are strictly monotone until a moment
$k_{0}(\beta)\leq\infty$, therefore the following limits exist

$\hat{c}(\beta)=\lim_{n\to\infty}c_{n}(\beta),\quad\hat{d}(\beta)=\lim_{n\to\infty}d_{n}(\beta).$
(2.9)

We can define the interval $(\hat{d}(\beta),\hat{c}(\beta))$, which includes
$(K_{1}-\delta,K_{3}+\delta)$ and is invariant under $G(\beta,\cdot)$. When
$k_{0}(\beta)=\infty$, $\\{\hat{d}(\beta),\hat{c}(\beta)\\}$ is a two-cycle.
Indeed, when (2.8) does not hold, we proceed to the limit, as $k\to\infty$, in
the equalities $G(\beta,d_{k}(\beta))=c_{k-1}(\beta)$ and
$G(\beta,c_{k}(\beta))=d_{k}(\beta)$ and get
$G(\beta,\hat{d}(\beta))=\hat{c}(\beta)$,
$G(\beta,\hat{c}(\beta))=\hat{d}(\beta).$ So, for this particular $\beta$ the
interval of the initial values with the desired convergence cannot be
increased. Moreover, the bound for control $\alpha$ is sharp: if it is
smaller, a cycle rather than an equilibrium can be an attractor.

All the above is summarized in the following lemma.

######  Lemma 2.7.

Let Assumption 1 hold, $\underline{\alpha}_{0}$ be defined as in (2.2),
sequences $(d_{n})_{n\in\mathbb{N}}$, $(c_{n})_{n\in\mathbb{N}}$ and numbers
$\underline{\kappa}$, $\bar{\kappa}$ by (2.6), and (2.5), $k_{0}$ by (2.7),
and $\beta\in\left(\underline{\alpha}_{0},\,1\right)$.

  1. (i)

For each $1\leq k=k(\beta)<k_{0}(\beta)\leq\infty$ we have:
$d_{k}(\beta)<x<K_{1}-\delta\implies G(\beta,x)<c_{k-1}(\beta)$ and
$c_{k}(\beta)>x>K_{3}+\delta\implies G(\beta,x)>d_{k}(\beta)$.

  2. (ii)

For each $1\leq k(\beta)<k_{0}(\beta)\leq\infty$ we have:
$G(\beta,\cdot):[d_{k}(\beta),c_{k-1}(\beta)]\to[d_{k}(\beta),c_{k-1}(\beta)]$.

  3. (iii)

The sequences $(c_{n}(\beta))_{n\in\mathbb{N}}$ and
$(d_{n}(\beta))_{n\in\mathbb{N}}$ are strictly monotone until the moment
$k_{0}(\beta)\leq\infty$,
$c_{n}(\beta)\uparrow\hat{c}(\beta)=\lim_{k\to\infty}c_{k}(\beta)$ and
$d_{n}(\beta)\downarrow\hat{d}(\beta)=\lim_{k\to\infty}d_{k}(\beta)$.

  4. (iv)

$\hat{c}(\beta)\in(K_{3}+\delta,\bar{\kappa}],\,\,\hat{d}(\beta)\in[\underline{\kappa},\,K_{1}-\delta)$
and
$G(\beta,\cdot):[\hat{d}(\beta),\hat{c}(\beta)]\to[\hat{d}(\beta),\hat{c}(\beta)]$.

  5. (v)

If (2.8) holds then sequences $(d_{n}(\beta))_{n\in\mathbb{N}}$ and
$(c_{n}(\beta))_{n\in\mathbb{N}}$ stop after either $k_{0}$ or $k_{0}+1$
steps, while if (2.8) fails then $\\{\hat{c}(\beta),\hat{d}(\beta)\\}$, as
defined in (2.9), is a two-cycle for the function $y=G(\beta,x)$.

  6. (vi)

For any $\tilde{\beta}_{1}>\tilde{\beta}_{2}>\underline{\alpha}_{0}$ and each
$k\in\mathbb{N}$, we have $d_{k}(\tilde{\beta}_{2})>d_{k}(\tilde{\beta}_{1})$
and $c_{k}(\tilde{\beta}_{1})>c_{k}(\tilde{\beta}_{2})$.

The next lemma is an extension of Lemma 2.5, it proves convergence of a
solution to an equilibrium when the initial value is in the interval dependent
on $\underline{\alpha}_{0}$, which is an extension of
$(K_{1}-\delta,K_{3}+\delta)$. Note that a solution to (1.2) converges to
$K_{0}$ or $K_{2}$ only in the cases of constant solutions with $x_{0}=K_{0}$
or $x_{0}=K_{2}$, respectively.

######  Lemma 2.8.

Let Assumption 1 hold, $\underline{\alpha}_{0}$ be defined as in (2.2),
$\alpha_{*}\in(\underline{\alpha}_{0},1)$, $\alpha^{*}\in(\alpha_{*},1)$,
$\alpha_{n}\in[\alpha_{*},\alpha^{*}]$ for all $n\in\mathbb{N}$. Let
$\hat{d}(\alpha_{*})$, $\hat{c}(\alpha_{*})$ be denoted as in (2.9). Then any
solution to (1.2) with $x_{0}\in(\hat{d}(\alpha_{*}),\hat{c}(\alpha_{*}))$
converges to an equilibrium.

Part (v) of Lemma 2.7 states that when $k_{0}(\alpha)=\infty$, the function
$G(\alpha,\cdot)$ has a two-cycle, so the condition
$\alpha\in(\underline{\alpha}_{0},1)$ might not be sufficient for convergence
of all solutions to an equilibrium. Example 3 in Section 4 illustrates this.
In order to fix this problem, we introduce a lower bound $\underline{\alpha}$
for $\alpha$ which can be larger than $\underline{\alpha}_{0}$

$\underline{\alpha}:=\inf\left\\{\beta\in(\underline{\alpha}_{0},1):\max_{x\in[K_{0},K_{1}]}G(\beta,x)<\hat{c}(\beta)~{}\mbox{or}\\!\\!\\!\\!\inf_{y\in(K_{3},g_{m}]}G(\beta,y)>\hat{d}(\beta)\right\\},$
(2.10)

where $\hat{c}(\beta)$ and $\hat{d}(\beta)$ are defined as in (2.9). To see
that the set in (2.10) is non-empty, we note first that
$\hat{c}(\beta)\in(K_{3},\bar{\kappa}],\,\,\hat{d}(\beta)\in[\underline{\kappa},\,K_{1})$,
see Lemma 2.7 (iv). When $\beta$ is close to 1,
$\max_{x\in[K_{0},K_{1}]}G(\beta,x)\approx K_{1}<K_{3}\leq\hat{c}(\beta)$ and
$\inf_{y\in(K_{3},g_{m})}G(\beta,y)\approx K_{3}>K_{1}\geq\hat{d}(\beta)$, so
such $\beta$ belongs to the set in the right-hand side of (2.10). Applying
Lemma 2.7 (vi), we conclude that each $\alpha\in(\underline{\alpha},1)$
belongs to the set inside of the braces in (2.10).

The following lemma highlights relations between $\underline{\alpha}_{0}$,
$k_{0}(\alpha)$ and $\underline{\alpha}$.

######  Lemma 2.9.

Let $\underline{\alpha}_{0}$, $k_{0}(\beta)$ and $\underline{\alpha}$ be
defined by (2.2), (2.7) and (2.10).

  1. (i)

$\underline{\alpha}_{0}=\underline{\alpha}$ if and only if for each
$\alpha>\underline{\alpha}_{0}$, condition (2.8) holds.

  2. (ii)

$\underline{\alpha}_{0}<\underline{\alpha}$ if and only if there exists
$\alpha>\underline{\alpha}_{0}$ such that condition (2.8) does not hold, i.e.
$k_{0}(\alpha)=\infty$.

We proceed now to the main result of this section.

######  Theorem 2.10.

Let Assumption 1 hold, $\underline{\alpha}$ be defined as in (2.10),
$\alpha_{*}\in(\underline{\alpha},1)$, $\alpha^{*}\in(\alpha_{*},1)$,
$\alpha_{n}\in[\alpha_{*},\alpha^{*}]$ for all $n\in\mathbb{N}$. Let $x$ be a
solution to (1.2) with $x_{0}\in(K_{0},\infty)$. Then $x$ converges to an
equilibrium.

######  Remark 2.

Based on results of Section 2.3, we conclude that $\underline{\alpha}$ defined
as in (2.10) is the best deterministic lower bound for the control, which
provides attraction of the solution to either $K_{1}$ or $K_{3}$. Even though
in general it is not so easy to find it, in some situations it is possible. In
Section 6.2 of the Appendix we demonstrate that when function $g$ is
differentiable outside of $[d_{1}(0),c_{1}(0)]$ we can find the lower
threshold $\underline{\alpha}_{1}$ which is calculated based on
$\underline{\alpha}_{0}$ and derivatives of $g$. Here $d_{1}(0)$ and
$c_{1}(0)$ are computed by (2.6) for $g(x)$ instead of $G(\beta,x)$ and
sometimes can be found easily.

###  2.4. Stochastically perturbed control

We start by introducing a complete filtered probability space
$(\Omega,{\mathcal{F}}$,
$\\{{\mathcal{F}}_{n}\\}_{n\in\mathbb{N}},{\mathbb{P}})$, where the filtration
$(\mathcal{F}_{n})_{n\in\mathbb{N}}$ is naturally generated by the sequence of
independent identically distributed random variables
$(\xi_{n})_{n\in\mathbb{N}}$, i.e.
$\mathcal{F}_{n}=\sigma\left\\{\xi_{1},\dots,\xi_{n}\right\\}$. The standard
abbreviation “a.s.” is further used for either “almost sure” or “almost
surely” with respect to a fixed probability measure $\mathbb{P}$, and “i.i.d.”
for “independent identically distributed”, to describe random variables. For
details of stochastic concepts and notations we refer the reader to
[MR1368405].

In many real-world models, in particular, in population dynamics, it is
natural to assume that noises are bounded, which we describe in the following
assumption, later a noise amplitude will be introduced.

######  Assumption 2.

$(\xi_{n})_{n\in\mathbb{N}}$ is a sequence of independent identically
distributed random variables satisfying $|\xi_{n}|\leq 1$, $\forall
n\in\mathbb{N}$. Moreover, for each $\varepsilon>0$,
$\mathbb{P}\\{\xi\in(1-\varepsilon,\,1]\\}>0$.

We consider a control perturbed by an additive noise,
$\alpha_{n}=\alpha+\ell\xi_{n+1}$,

$x_{n+1}=g(x_{n})-(\alpha+\ell\xi_{n+1})(g(x_{n})-x_{n}),\quad
x_{0}\in\mathbb{R},\quad n\in{\mathbb{N}}_{0}.$ (2.11)

Here $\alpha\in(0,1)$, random variables $\xi_{n}$ satisfy Assumption 2,
$\ell>0$ is a noise amplitude.

Let $\underline{\alpha}_{0}$ and $\underline{\alpha}$ be defined as in (2.2)
and (2.10), respectively. In this section we decrease the lower bound
$\underline{\alpha}$ for a control parameter $\alpha$ proposed in the previous
sections, applying stochastic perturbations. Set

$\begin{split}&\alpha_{n}=\alpha+\ell\xi_{n},\quad\alpha\in\bigl{(}0.5(\underline{\alpha}_{0}+\underline{\alpha}),\,\,\underline{\alpha}\bigr{)},\quad\ell\in\bigl{(}\underline{\alpha}-\alpha,\,\,\min\\{1-\alpha,\alpha-\underline{\alpha}_{0}\\}\bigr{)}.\end{split}$
(2.12)

Since $\underline{\alpha}\geq\underline{\alpha}_{0}$, see Lemma 2.9, any
$\alpha$ satisfying (2.12) will be smaller than $\underline{\alpha}$ which is
the best lower estimate for the deterministic case.

The following lemma was proved in [MR3499497, MR3543576] and is a corollary of
the Borel-Cantelli Lemma.

######  Lemma 2.11.

Let sequence $(\xi_{n})_{n\in\mathbb{N}}$ satisfy Assumption 2. Then, for each
nonrandom $\mathbf{J}\in\mathbb{N}$, $\varepsilon>0$ and a random moment
$\mathcal{M}$, there is a random moment
$\mathcal{N}=\mathcal{N}(\mathbf{J},\varepsilon,\mathcal{M})\geq{\mathcal{M}}$,
$\mathcal{N}\in{\mathbb{N}}$, such that $\mathbb{P}\\{\xi_{\mathcal{N}+i}\geq
1-\varepsilon,\,\,i=0,1,\dots,\mathbf{J}\\}=1.$

######  Theorem 2.12.

Let Assumptions 1 and 2 hold, $\underline{\alpha}_{0}$ and
$\underline{\alpha}$ be defined as in (2.2) and (2.10), $\alpha$ and $\ell$
satisfy (2.12). Then a solution $x$ to (2.11) with $x_{0}\in(K_{0},\infty)$
converges to one of the equilibrium points with the total probability one.

##  3\. An arbitrary number of equilibrium points

In this section we consider a continuous function $g$ which might have more
than four equilibrium points, see the fourth iterate of Ricker’s map in Fig. 2
as an example. An approach similar to previous sections is applicable, however
it becomes quite technical, therefore we reduce considerations to the analogue
of Assumption 1 when $\delta=0$. The general case we leave for the future
research.

Figure 2. The fourth iteration of the Ricker map for $r=2.6$.

We aim to stabilize every second point only, so we define the number of these
points as

$i_{0}:=\left[\frac{j_{0}}{2}-1\right],\quad
I:=\left\\{0,1,\dots,i_{0}\right\\},$ (3.1)

where $\left[t\right]$ is the largest integer not exceeding $t$.

######  Assumption 3.

Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous function such that

$\begin{split}&g(K_{j})=K_{j}\in\mathbb{K}:=\\{K_{j},\,j=0,1,2,\dots,j_{0}\\},\,\,K_{j_{0}}\leq\infty,\\\
&g(x)-x\geq 0,\,\mbox{for}\,\,x\in(K_{0},K_{1}),\\\ &g(x)-x\mbox{ alternates
its sign on adjacent }\,(K_{j},K_{j+1}),\\\
&\,j=0,1,2,\dots,j_{0}-1,\,\,\mbox{$g(x)-x$ can vanish for some
$x\in(K_{j},K_{j+1})$},\\\ &x\geq g(x)\geq b,\,\,\mbox{for
some}\,\,b\in\mathbb{R},\,\,\mbox{when}\,\,x\in(K_{j_{0}-1},\infty),\\\
&\mbox{if}\,\,K_{j_{0}}=\infty,\,j_{0}\,\mbox{is even}.\end{split}$ (3.2)

The last two lines in (3.2) correspond to higher death rates for
overpopulation, which is satisfied for any iterates of the Ricker, logistic
and other population dynamics maps. In the chaotic case, these maps are
characterized by much higher global Lipschitz constants on one of two adjacent
segments with a common potentially stabilizable point, which corresponds to a
one-side Lipschitz condition at each such point in Assumption 3,

$\begin{split}&K_{2i+1}-g(x)\leq
L^{+}_{2i+1}(x-K_{2i+1}),~{}x\in(K_{2i+1},K_{2i+2}),~{}i\in I,\\\
&g(x)-K_{2i+1}\leq
L^{-}_{2i+1}(K_{2i+1}-x),~{}x\in(K_{2i},K_{2i+1}),~{}i=0,1,\dots,\left[\frac{j_{0}-1}{2}\right].\end{split}$
(3.3)

An efficient control is possible, only if at least one of one-side Lipschitz
constants at each point $K_{2i+1}$ is finite.

######  Assumption 4.

Let $g$ from Assumption 3 satisfy that either $j_{0}$ is even, or $g$ is left
locally Lipschitz at $K_{j_{0}}<\infty$, and
$\min\left\\{L^{-}_{2i+1},\,L^{+}_{2i+1}\right\\}<\infty$, $i\in I$, where
$L^{+}_{2i+1},L^{-}_{2i+1}\in(0,\infty]$ are defined in (3.3).

Under Assumptions 3 and 4, for $K_{j_{0}}<\infty$, either $j_{0}$ is even, or
$g$ is left locally Lipschitz at $K_{j_{0}}$. In the former case, $g(x)<x$ for
$x\in(K_{j_{0}-1},K_{j_{0}})$, and, due to continuity of $g$, the function
$(g(x)-K_{j_{0}})/(K_{j_{0}}-x)$ is continuous on $[K_{0},K_{j_{0}-1}]$ and

$L^{-}_{j_{0}}:=\max_{x\in[K_{0},K_{j_{0}}]}\frac{g(x)-K_{j_{0}}}{K_{j_{0}}-x}=\max_{x\in[K_{0},K_{j_{0}-1}]}\frac{g(x)-K_{j_{0}}}{K_{j_{0}}-x}<\infty.$

In the latter case, if $L^{*}$ is a local Lipschitz left constant,
$g(x)-K_{j_{0}}\leq L^{*}(K_{j_{0}}-x)$,
$x\in(K_{j_{0}}-\varepsilon,K_{j_{0}})$, we again use continuity of
$(g(x)-K_{j_{0}})/(K_{j_{0}}-x)$ on $[K_{0},K_{j_{0}}-\varepsilon]$ to get

$L^{-}_{j_{0}}:=\max_{x\in[K_{0},K_{j_{0}}]}\frac{g(x)-K_{j_{0}}}{K_{j_{0}}-x}=\max\left\\{L^{*},\max_{x\in[K_{0},K_{j_{0}}-\varepsilon]}\frac{g(x)-K_{j_{0}}}{K_{j_{0}}-x}\right\\}<\infty.$

For $K_{0}$, $g(x)>x\geq K_{0}$, $x\in[K_{0},K_{1}]$, the function
$(g(x)-K_{0})/(x-K_{0})$ is continuous on $[K_{1},K_{j_{0}}]$, therefore

$L^{+}_{0}:=-\min_{x\in[K_{0},K_{j_{0}}]}\frac{g(x)-K_{0}}{x-K_{0}}=-\min_{x\in[K_{1},K_{j_{0}}]}\frac{g(x)-K_{0}}{x-K_{0}}<\infty.$

Summarizing, under Assumptions 3 and 4, when $K_{j_{0}}<\infty$, there exist
finite positive constants $L^{-}_{j_{0}}$ and $L^{+}_{0}$ such that

$\begin{split}&g(x)-K_{j_{0}}\leq L^{-}_{j_{0}}(K_{j_{0}}-x),\quad
x\in(K_{0},K_{j_{0}}),\quad L^{-}_{j_{0}}<\infty,\\\
&g(x)-K_{0}\geq-L^{+}_{0}(x-K_{0}),\quad x\in(K_{0},K_{j_{0}}),\quad
L^{+}_{0}<\infty.\end{split}$ (3.4)

######  Remark 3.

Once we are free to choose $j_{0}$ in the control scheme, the condition
concerning the value of $j_{0}$ in Assumption 4 is not really a limitation,
assuming local one-side Lipschitz condition at each $K_{2i+1}$ is sufficient.
Whenever $g:(K_{0},\,K_{j_{0}})\to(K_{0},\,K_{j_{0}})$, both inequalities in
(3.4) hold with arbitrarily small constants $L^{-}_{j_{0}}$ and $L^{-}_{0}$.

Denote

$L_{2i+1}:=\min\left\\{L^{-}_{2i+1},\,L^{+}_{2i+1}\right\\},~{}i\in
I,\quad\bar{L}:=\max\left\\{L^{+}_{0},\,L^{-}_{j_{0}},\,L_{2i+1},\,i\in
I\right\\},$ (3.5)

where the notations in (3.3) and (3.4) are used. Without loss of generality,
we assume that $L>1$. Now, introduce the sets

$\begin{split}I^{+}:=\left\\{i\in I:L_{2i+1}=L^{+}_{2i+1}\right\\},\quad
I^{-}:=\left\\{i\in I:L_{2i+1}=L^{-}_{2i+1}<L^{+}_{2i+1}\right\\},\end{split}$
(3.6)

i.e. for $i\in I^{+}$ we have $L^{+}_{2i+1}\leq L^{-}_{2i+1}$, and
$L^{+}_{2i+1}>L^{-}_{2i+1}$ for $I^{-}$. Also, $I^{+}\cup I^{-}=I$ and
$I^{+}\cap I^{-}=\emptyset$.

Similarly to Lemma 2.3, we can justify the following result.

######  Lemma 3.1.

Let $g$ from Assumption 3 satisfy conditions (3.3) and (3.4), $G$ be defined
as in (2.1).

  1. (i)

For
$a:=\max\left\\{L^{-}_{j_{0}}/(1+L^{-}_{j_{0}}),\,\,L^{+}_{0}/(1+L^{+}_{0})\right\\}$,
$\alpha\in(a,1]$, we have
$G(\alpha,\cdot):(K_{0},K_{j_{0}})\to(K_{0},K_{j_{0}}).$

  2. (ii)

Let $\alpha\in\bigl{(}\bar{L}/(\bar{L}+1),1\bigr{)}$, and $x_{n}$ be a
solution to (1.2) with $\alpha_{n}\in(\bar{L}/(\bar{L}+1),1)$.

    1. (a)

If $x_{0}\in(K_{2i},K_{2i+1})$, $i\in I^{-}$ then
$G(\alpha,x_{0})\in(K_{2i},K_{2i+1})$, and $x_{n}$ is non-decreasing and
converges either to $K_{2i+1}$ or to some equilibrium in $(K_{2i},K_{2i+1})$.

    2. (b)

If $x_{0}\in(K_{2i+1},K_{2i+2})$, $i\in I^{+}$ then
$G(\alpha,x_{0})\in(K_{2i+1},K_{2i+2})$, and $x_{n}$ is non-increasing and
converges either to $K_{2i+1}$ or to some equilibrium in $(K_{2i},K_{2i+1})$.

    3. (c)

If $i\in I^{+}$ and $i+1\in I^{-}$, then
$G(\alpha,\cdot):[K_{2i+1},K_{2i+3}]\to[K_{2i+1},K_{2i+3}]$. Moreover, if
$x_{0}\in(K_{2i},K_{2i+2})$ then $x_{n}$ converges to either $K_{2i+1}$, or
$K_{2i+3}$, or an equilibrium in $(K_{2i+1},K_{2i+2})\cup(K_{2i+2},K_{2i+3})$
and is monotone.

Everywhere in this paper we assume $\alpha_{n}>\bar{L}/(\bar{L}+1)$, which
makes it possible to apply Lemma 3.1. Definition of sets in (3.6) allows us to
distinguish between $K_{2j+1}$ depending on the side where the Lipschitz
constant is smaller, i.e. whether $L_{2i+1}=L^{+}_{2i+1}$ or
$L_{2i+1}=L^{-}_{2i+1}$. Similarly to the four-equilibrium case, we can
establish stability of equilibria $K_{2j+1}\in\mathbb{K}$ with odd indexes
only. Lemma 3.1 describes the behavior of a solution when
$x_{0}\in(K_{2i},K_{2i+1})$, $i\in I^{-}$, and $x_{0}\in(K_{2i+1},K_{2i+2})$,
$i\in I^{+}$. However, when $x_{0}\in(K_{2i},K_{2i+1})$, $i\in I^{+}$, and
$x_{0}\in(K_{2i+1},K_{2i+2})$, $i\in I^{-}$, the situation is different: the
solution can get out of the interval, start traveling from one interval to
another including infinite circulation between some intervals.

The general idea contains packing the intervals surrounding a potentially
stabilizable equilibrium into blocks, based on whether the left or the right
Lipschitz constant is bigger. At the very beginning, we apply a control which
makes open segments with a smaller constant monotone attractors. Thus, if we
have a group $\mathcal{V}_{0}$ of consecutive segments with prevailing right
constants, a solution with an initial point in this group can only be
switching to the segments to the left, and therefore converges to one of the
equilibrium points in $\mathcal{V}_{0}$. Similarly, for a group
$\tilde{\mathcal{V}}$ of consecutive segments with prevailing left constants,
any solution with an initial point in this group can only be switching to the
segments to the right, and therefore converges to one of the equilibrium
points in $\tilde{\mathcal{V}}$. Our main analysis is devoted to the block
$\mathcal{V}_{2s+1}$, $s=0,\dots,s_{0}-1$, consisting of two back-to-back
groups of both types, between which circulation is possible. We introduce two-
step control process: first keeping all solutions inside a designated block,
and then eliminating circulation between two groups inside each block. For
brief illustration of intervals we refer to Fig. 2, where we have
$L_{1}^{-}>L_{1}^{+}$, $L_{3}^{-}<L_{3}^{+}$, $L_{5}^{-}>L_{5}^{+}$,
$L_{7}^{-}\\!<\\!L_{7}^{+}$, $\tilde{\mathcal{V}}=\mathcal{V}_{0}=\emptyset$,
$\mathcal{V}_{1}=\\{(K_{0},K_{1}),\,(K_{1},K_{2}),\,(K_{2},K_{3}),\,(K_{3},K_{4})\\}$,
$\mathcal{V}_{3}=\\{(K_{4},K_{5}),\,(K_{5},K_{6}),\,(K_{6},K_{7}),\,(K_{7},\infty)\\}$.

The next theorem is the main result for the deterministic control on all
$(K_{0},K_{j_{0}})$, when only one-side Lipschitz constants at odd-numbered
equilibrium points are supposed to be finite. Most details on classification
of the intervals along with the proof of Theorem 3.2 are deferred to the
Appendix. Note that in Theorem 3.2 constant solutions with $x_{0}=K_{2j}$ are
excluded from consideration, see the remark before Lemma 2.8.

######  Theorem 3.2.

Let Assumptions 3 and 4 hold, $I$ be defined as in (3.1). Then there exists a
control $\alpha_{*}\in\bigl{(}\bar{L}/(\bar{L}+1),1\bigr{)}$ such that for
each $\alpha^{*}\in(\alpha_{*},1)$, if $\alpha_{n}\in[\alpha_{*},\alpha^{*}]$
for all $n\in\mathbb{N}$, a solution $x$ to (1.2) with
$x_{0}\in(K_{0},K_{j_{0}})$ converges either to some $K_{2i+1}$, $i\in I$, or
to any equilibrium inside $(K_{p},K_{p+1})$, $0\leq p\leq j_{0}-1$.

######  Remark 4.

The results of Theorem 3.2 can be extended to an infinite number of
equilibrium points, $\mathbb{K}=\\{K_{i},\,i\in\mathbb{N}\\}$ with
$\lim_{i\to\infty}K_{i}=\infty$, if we consider a finite subinterval. For each
$p\in\mathbb{N}$ and $x_{0}\in(K_{0},K_{p})$, Theorem 3.2 holds with some
$\alpha=\alpha(p)$.

In some situations, when a control is perturbed by an additive noise,
$\alpha+\ell\xi_{n}$, Theorem 3.2 can be improved by decreasing the lower
bound on $\alpha$. However, in the case of an arbitrary number of equilibrium
points, even the statement of all necessary conditions is quite involved, so
we defer most of this part to the Appendix.

##  4\. Examples and simulations

In this section we consider three examples illustrating our results. In
particular, simulations demonstrate that introduction of noise into a
deterministic control sometimes extends the range of $\alpha$ which guarantees
stability.

######  Example 1.

Consider the second iteration of the Ricker map $g=f^{2}$, $r=2.7$, as
illustrated by Fig. 1. The function $g=f^{2}$ is infinitely differentiable and
has 3 positive fixed points $K_{1}\approx 0.214$, $K_{2}=1$, $K_{3}\approx
1.786$. The map $g=f^{2}$ has a minimum at the point $\frac{1}{r}\approx
0.37\in(K_{1},K_{2})$ with the value $g_{\min}\approx 0.142$, and monotone
derivatives on each intervals, $(K_{1},K_{2})$ and $(K_{2},K_{3})$, and it has
two maximums with the same value $g_{\max}=\frac{e^{r-1}}{r}\approx 2.027$. We
estimate Lipschitz constant $L$ on these intervals as $L\approx 1.65$. The
left Lipschitz constant on $(K_{0},K_{1})=(0,0.214)$ is quite large, it
exceeds 9.8, leading to $(L-1)/(L+1)>0.8$. Thus application of results of
[MR2966855, Theorem 2.5] gives us a lower estimate for $\alpha_{n}$ exceeding
0.8, taking into account that the right Lipschitz constant only gives
$(L-1)/(L+1)\approx 0.2452$.

We show that $\beta\approx 0.249$ belongs to the right-hand side of (2.2) for
some $\delta>0$. The corresponding function $G(\beta,\cdot)$ takes its minimum
on $(K_{1},K_{2})$ at $x_{\min~{}\beta}$, which can be found from the equation
$g^{\prime}(x_{\min\beta})=-\beta/(1-\beta)$. Approximating $g^{\prime}(x)$
numerically and taking into account $g^{\prime}(x_{\min\beta})\approx-0.331$,
we get $x_{\min~{}0.249}\approx 0.3387$.

For $x\in(K_{1},K_{2})$ and $\beta=0.249$, we estimate

$\displaystyle G(\beta,x)\geq(1-\beta)g_{\min}+\beta
x_{\min\,\beta}>-\delta+K_{1},$
$\displaystyle\delta>K_{1}-(1-\beta)g_{\min}-\beta x_{\min\,\beta}\approx
0.022,$

which is consistent with the graph on Fig. 1. So the inequality on the second
line of (2.2) holds. Since $|G^{\prime}(\beta,x)|$ is decreasing for
$x>K_{3}$, we do not need to take care about the inequality on the third line
of (2.2).

For $\beta=0.249$ we have
$\max_{x\in(0,K_{1})}G(\beta,x)\leq(1-\beta)\max_{x\in(0,K_{1})}g(x)+\beta
K_{1}\leq 1.786=K_{3}<K_{3}+\delta$, so, by (2.6), (2.7), we have
$d_{1}(\beta)=d_{0}(\beta)=\hat{d}(\beta)=K_{1}-\delta$, $k_{0}(\beta)=1$,
and, by (2.10) we have $\beta>\underline{\alpha}$. Thus, Theorem 2.10 implies
attractivity of $K_{1}$ and $K_{3}$.

A bifurcation diagram on Fig. 3 (left) confirms the result.

Figure 3. A bifurcation diagram for the second iterate of the Ricker map with
$r=2.7$, $\alpha\in(0.1,0.3)$ and (left) no noise, (right) $\ell=0.15$.

When a control is stochastically perturbed by the noise with the amplitude
$\ell=0.15$, the bifurcation diagram in Fig. 3 (right) demonstrates that
stabilization is achieved at $\alpha\approx 0.215$, which is smaller than for
the deterministic case $\alpha\approx 0.235$, see Fig. 3 (left), and is
aligned with the result of Theorem 2.12.

######  Example 2.

Consider the third iteration of the Ricker map $g=f^{3}$, $r=3.5$, which has 7
equilibrium points, $K_{i}$, $K_{8}:=\infty$, $|g^{\prime}(K_{2i+1})|\leq
15.62=L$, $i=0,1,\dots,7$, and the right Lipschitz constants $L^{+}_{2i+1}$ on
the intervals $(K_{2i+1},K_{2i+2})$, $i=0,\dots 3$ do not exceed $L\approx
15.62$. However, some of left Lipschitz constants $L^{-}_{2i+1}$ on the
intervals $(K_{2i},K_{2i+1})$ are greater than 300.

Applying the comments from Section 3 and classification from Section 6.3.1, we
conclude that all the intervals $(K_{p},K_{p+1})$, $p=1,2,\dots,7$ belong to
the block $\tilde{\mathcal{V}}$. So for $\alpha_{n}\geq 0.94$, the circulation
of solution between intervals is impossible, and all the equilibria
$K_{2i+1}$, $i=0,\dots,3$ are stable. The bifurcation diagram in Fig. 4 (left)
demonstrates a slightly smaller value of $\approx 0.88$. And again, this
number is reduced to $\approx 0.86$ if a noise with $\ell=0.06$ is introduced,
see Fig. 4 (right).

Figure 4. A bifurcation diagram for the third iterate of the Ricker map with
$r=3.5$ for $\alpha\in(0.75,0.9)$ and (left) without noise, (right)
$\ell=0.06$. The last bifurcation leading to two stable equilibrium points
occurs for $\alpha\approx 0.88$ in the deterministic case and $\alpha<0.86$ in
the stochastic case.

######  Example 3.

This example illustrates the results of Section 3. Define

$g(x)=\left\\{\begin{array}[]{cccc}0,&x\in(-\infty,0],\vspace{2mm}\\\
\frac{163}{63}x,&x\in(0,\,0.9],\vspace{2mm}\\\
x+\frac{0.6}{0.42\sqrt{0.1}}\sqrt{1-x},&\quad x\in(0.9,\,1],\vspace{2mm}\\\
x-\frac{5}{23\pi}\sin[10\pi(x-1)],&x\in(1,1.2],\vspace{2mm}\\\
x-\frac{0.6}{0.42\sqrt{0.3}}\sqrt{x-1.2},&\quad
x\in\left(1.2,\,1.5\right],\vspace{2mm}\\\ 2x-\frac{41}{14},&\quad
x\in\left(1.5,\frac{41}{14}\right],\vspace{2mm}\\\
x+10\sqrt{(x-\frac{41}{14})(3-x)},&\quad
x\in\left(\frac{41}{14},3\right],\vspace{2mm}\\\ x-\sqrt{x-3},&\quad
x\in(3,\infty),\end{array}\right.$ (4.1)

which has 6 equilibrium points, see Fig. 5.

Figure 5. The graph of the map defined in (4.1), together with $y=x$.

Here both left- and right-side derivatives at $K=3$ are infinite, so for
efficient control, we have to reduce ourselves to the segment $[0,41/14]$
getting the set of equilibrium points
$\mathbb{K}=\left\\{0,1,\,1.1,\,1.2,\,41/14\right\\}$, $K_{4}=41/14<\infty$.
The equilibrium points $K_{1}=1$ and $K_{3}=1.2$ are unstable, since
$g^{\prime}(1^{-})=-\infty=g^{\prime}(1.2^{+})$ and
$g^{\prime}(1^{+})=-27/23=L=g^{\prime}(1.2^{-})$, so
$\beta_{0}=\underline{\alpha}=L/(1+L)=0.54$. By (4.1), for each $\beta$, the
point of minimum of $G(\beta,x)$ is $\underline{\kappa}=0.9$, and the point of
maximum is $\bar{\kappa}=1.5$ and $g:[0,\,41/14]\to[0,\,41/14]$.

By straightforward calculation we show that for each
$\beta\in(0.54,\,0.6040)$, the function $G(\beta,x)$ has two unstable
2-cycles, located in $[0.9;\,1)$ and $[1.2,\,1.5)$, respectively. For example,
for $G(0.58,x)$, there are two two-cycles at approximately $\\{0.9,\,1.5\\}$
and $\\{0.9749,\,1.2755\\}$. For $d_{k}$, $\hat{d}$, $c_{k}$, $\hat{c}$
defined by (2.6) and (2.9) (when $\beta=0.58$) we have
$0.9<0.9749\leq\hat{d}<d_{k}<K_{1}=1$, $1.2=K_{3}<c_{k}<\hat{c}\leq
1.2755<1.5$, $k\in\mathbb{N}$, so the equilibrium points $1$ and $1.2$ are
unstable. For $\beta=0.6040$ there is only one 2-cycle,
$\approx\\{0.95,\,1.3497\\}$. For $\beta>0.6040$ there is no two-cycle.

Fig. 6 (left) presents a bifurcation diagram for $G(\alpha,\cdot)$ with $g$
defined in (4.1) and demonstrates efficient stabilization in the chosen
segment $[0,\,41/14]$ and attractors in addition to the two equilibrium points
for $\alpha<0.605$, as the theory predicts. Fig. 6 (right) contains a
bifurcation diagram for $G(\alpha,\cdot)$, when the control $\alpha$ is
perturbed by the Bernoulli noise with $\ell=0.04$, and two stable equilibrium
points starting with a smaller $\alpha\approx 0.535$.

Figure 6. A bifurcation diagram for the map defined in (4.1) with
$\alpha\in(0.45,0.65)$ and (left) no noise, we have two stable equilibrium
points starting from $\alpha\approx 0.605$, (right) for Bernoulli noise with
$\ell=0.04$, the last bifurcation happens for smaller $\alpha\approx 0.535$.
Here the two attractors correspond to two stable equilibrium points with
separate basins of attraction, not to a cycle.

##  5\. Discussion and future research

For stabilization of maps with PBC, if a map is unimodal with a negative
Schwarzian derivative, the best control bound is easily computed using the
derivative at an equilibrium [MR2741904]. If a map is not unimodal and has
several fixed points, global Lipschitz two-side constants can be used to
compute a required control bound in a similar way [MR2966855]. However, for
very large one-side Lipschitz constants, this bound is quite close to one and
is far from being optimal. When the difference between two one-side constants
is significant, it may be preferable to choose a control in such a way that,
once a point gets into a smoother area, it stays there forever. Unfortunately,
for models with many fixed points, in particular, for iterates of common maps,
this approach does not guarantee stabilization, as a solution can wander
between “steep” areas. The algorithm to choose an optimal control bound,
together with making it lower in the stochastic case, is the main
accomplishment of the present paper. Specifically, we managed

  * •

to overcome the problem of large global Lipschitz constants compared to a
derivative at an equilibrium by considering one-side constants only, which
allows to relax control conditions;

  * •

to include the case when a one-side derivative is infinite (in this case the
constant $L/(L+1)$ is sharp);

  * •

introducing a noise in the control, to relax restrictions on the average
control intensity even more.

PBC was proved to be an efficient tool in stabilizing multiple equilibrium
points, even in the case when the continuous map is not smooth. Introduction
of noise allowed to lower the level of average control, and this result
complements [MR3499497]. Compared to [MR4145192], the results of the present
paper are global, not local, which gives an advantage in practical
implementations.

Some possible extensions of the present work are listed below.

  * •

Section 2 can be extended to the case of $\delta>0$ and an arbitrary number of
equilibrium points. Also, the results for iterates of maps can be stated in
the terms of the original function and a pulse PBC control.

  * •

It would be interesting to generalize our results to systems of difference
equations subject to PBC control [MR3166513], for example, relating to
[MR3166513] in a stochastic case and combining [MR3551680, MR3681357] with
PBC-type control.

  * •

Following the previous item, results for systems can be readily applied to
models of population dynamics, controlling competition outcomes. The essential
role of stochasticity was recently analyzed in [MR3975878, MR3700056], see
also references therein.

## Acknowledgments

The authors are very grateful to anonymous reviewers whose thoughtful comments
significantly contributed to the quality of presentation of our results.

## References

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##  6\. Appendix

###  6.1. Proofs for sections 2

####  6.1.1. Proof of Lemma 2.1

We have
$x_{n+1}-x_{n}=(1-\alpha_{n+1})g(x_{n})-(1-\alpha_{n+1})x_{n}=(1-\alpha_{n+1})(g(x_{n})-x_{n}).$
So
$|g(x_{n})-x_{n}|=|x_{n+1}-x_{n}|/(1-\alpha_{n+1})\leq|x_{n+1}-x_{n}|/(1-a)\to
0,\quad\text{as}\quad m\to\infty,$ and, by continuity of $g$, we have
$g(x^{*})-x^{*}=\lim_{n\to\infty}[g(x_{n})-x_{n}]=0.$

####  6.1.2. Proof of Lemma 2.2

Parts (i)-(iii) are immediate results of definition (2.1) and continuity of
$G$. To prove (iv), we note that since
$1-\hat{\mu}=1-\frac{\mu-\mu_{0}}{1-\mu_{0}}=\frac{1-\mu}{1-\mu_{0}}$, we have
$(1-\hat{\mu})[(1-\mu_{0})=1-\mu$ and $\hat{\mu}=\mu-(1-\hat{\mu})\mu_{0}$.
Therefore,

$\displaystyle G(\mu,x)$ $\displaystyle=(1-\hat{\mu})(1-\mu_{0})g(x)+\mu
x=(1-\hat{\mu})G(\mu_{0},x)-(1-\hat{\mu})\mu_{0}x+\mu x$
$\displaystyle=(1-\hat{\mu})G(\mu_{0},x)+[\mu-(1-\hat{\mu})\mu_{0}]x=(1-\hat{\mu})G(\mu_{0},x)+\hat{\mu}x.$

####  6.1.3. Proof of Lemma 2.3

Note that $\alpha>L/(L+1)\implies L(1-\alpha)-\alpha<0$.

(i) For $x\in(K_{1},K_{2})$, by Assumption 1, we get

$\displaystyle K_{2}$
$\displaystyle>x>G(\alpha,x)=(1-\alpha)(g(x)-K_{1})+\alpha(x-K_{1})+K_{1}$
$\displaystyle\geq-L(1-\alpha)(x-K_{1})+\alpha(x-K_{1})+K_{1}$
$\displaystyle=-[L(1-\alpha)-\alpha](x-K_{1})+K_{1}>K_{1}.$

The case $x\in(K_{2},K_{3})$ is similar.

(ii) For $x\in(K_{1}-\delta,K_{1})$, we have $K_{1}-\delta<x<G(\alpha,x)$ and

$G(\alpha,x)=(1-\alpha)(g(x)-K_{1})+\alpha(x-K_{1})+K_{1}\leq[L(1-\alpha)-\alpha](K_{1}-x)+K_{1}<K_{1}.$

The case $x\in(K_{3},K_{3}+\delta)$ is similar.

(iii) Parts (i) and (ii) imply that the solution $x_{n}$ remains in the same
interval (either $(K_{1}-\delta,K_{1})$ or $(K_{1},K_{3})$ or
$(K_{3},K_{3}+\delta)$) as its initial value $x_{0}$. In each of the three
intervals the sequence $(x_{n})_{n\in\mathbb{N}}$ is monotone and bounded, and
therefore is convergent. The application of Lemma 2.1 completes the proof.

####  6.1.4. Proof of Lemma 2.4

Take some $\alpha>L/(L+1)$. Applying Lemma 2.3 (i)-(ii) we conclude that
$\min_{x\in[K_{1}-\delta,K_{1}]}G(\beta,x)>K_{1}-\delta$ and
$\min_{x\in[K_{1},K_{2}]}G(\beta,x)>K_{1}.$ Similar estimations can be done
for $\max_{x\in[K_{2},K_{3}+\delta]}G(\beta,x)$. So $\alpha$ belongs to the
set on the second and third lines of (2.2), which, therefore, is non-empty. It
also implies $\alpha\geq\underline{\alpha}_{0}$, and proves Part (ii).

####  6.1.5. Proof of Lemma 2.5

The case $\delta=0$ is covered by Lemma 2.3 (i),(iii). So we consider only
$\delta>0$.

(i) Let $x\in(K_{1}-\delta,K_{2})$, then $G(\alpha,x)>K_{1}-\delta$. If
$x\in(K_{1},K_{2})$ then $G(\alpha,x)<x<K_{2}$ by (2.2). For
$x\in(K_{1}-\delta,K_{1})$, since $\delta<K_{2}-K_{1}$, we have

$\begin{split}G(\alpha,x)&=(1-\alpha)(g(x)-K_{1})+\alpha(x-K_{1})+K_{1}\\\
&\leq L(1-\alpha)(K_{1}-x)+\alpha(x-K_{1})+K_{1}\\\
&=[L(1-\alpha)-\alpha](K_{1}-x)+K_{1}<(x-K_{1})+K_{1}<\delta+K_{1}<K_{2}.\end{split}$
(6.1)

The case $x\in(K_{2},K_{3}+\delta)$ is similar.

(ii) Note that when $\underline{\alpha}_{0}=L/(L+1)$, we are under the
assumptions of Lemma 2.3. Assume that $\underline{\alpha}_{0}<L/(L+1)$. If
$\alpha_{*}<L/(L+1)$, we set $q:=L(1-\alpha_{*})-\alpha_{*}$ and note that
$q\in(0,1)$ and $L(1-\alpha_{n})-\alpha_{n}<q$. Let
$x_{n}\in(K_{1}-\delta,K_{1})$. Estimation (6.1) gives us
$G(\alpha_{n+1},x_{n})-K_{1}<[L(1-\alpha_{n+1})-\alpha_{n+1}](K_{1}-x_{n})<q(K_{1}-x_{n}).$
So either $G(\alpha_{n+1})$ remains in $(K_{1}-\delta,K_{1})$ after some
moment or it changes the position around $K_{1}$ infinitely many times. In
both cases $\lim_{n\to\infty}x_{n}=K_{1}$. If $\alpha_{*}=L/(L+1)$, we set
$\underline{\alpha}_{*}:=0.5[\underline{\alpha}_{0}+L/(L+1)]\in\left(\underline{\alpha}_{0},L/(L+1)\right)$
and define $q:=L(1-\underline{\alpha}_{*})-\underline{\alpha}_{*}$. Since
$\alpha_{n}\geq\alpha_{*}=L/(L+1)>\underline{\alpha}_{*}$, the rest of the
proof is the same as above.

####  6.1.6. Proof of Lemma 2.6

We need to check only the case
$x_{0}\in(K_{0},K_{1}-\delta)\cup(K_{3}+\delta,\infty)$. Let
$x_{0}\in(K_{0},K_{1}-\delta)$, the other case is similar. If the first
relation in (2.3) holds, $x_{n}$ cannot remain in $(K_{0},K_{1}-\delta)$
forever, since in that case it should converge to the equilibrium which is not
in $(K_{0},K_{1}-\delta)$. So it enters $(K_{1}-\delta,K_{3}+\delta)$ in a
finite number of steps, and we can apply Lemma 2.5. If the second relation in
(2.3) holds then $x_{\bar{n}}$ can get out to $(K_{3}+\delta,\infty)$ for some
$\bar{n}\in\mathbb{N}$. In this case $x_{n}\in(K_{1}-\delta,x_{\bar{n}})$ for
all $n\geq\bar{n}$, so $x_{n_{0}}\in(K_{1}-\delta,K_{3}+\delta)$ for some
$n_{0}\geq\bar{n}$, and we can apply Lemma 2.5 again.

####  6.1.7. Proof of Lemma 2.7

Parts (i)-(ii) follow from definitions (2.6) and the fact that $G(\beta,x)\geq
x$ for $x\in(K_{0},K_{1})$, and $G(\beta,x)\leq x$ for $x\in(K_{3},g_{m})$,
where $g_{m}$ is defined in (2.4). Definitions (2.6), Part (i) and Lemma 2.3
imply Part (iii). Part (iv) follows from (2.6), definitions (2.5) of
$\underline{\kappa}(\beta)$, $\bar{\kappa}(\beta)$ and Part (ii).

The first part of (v) follows from definition (2.7). When (2.8) does not hold,
we proceed to the limit as $k\to\infty$ in the equalities
$G(\beta,d_{k}(\beta))=c_{k-1}(\beta)$ and
$G(\beta,c_{k}(\beta))=d_{k}(\beta)$ and get
$G(\beta,\hat{d}(\beta))=\hat{c}(\beta)$,
$G(\beta,\hat{c}(\beta))=\hat{d}(\beta)$ by continuity.

To prove (vi), for $\tilde{\beta}_{1}>\tilde{\beta}_{2}$, we use that
$G(\tilde{\beta}_{2},x)>G(\tilde{\beta}_{1},x)$ for $x\in(K_{0},K_{1})$, and
$G(\tilde{\beta}_{2},x)<G(\tilde{\beta}_{1},x)$ for $x\in(K_{3},g_{m})$.
Applying (2.6) we show that
$d_{1}(\tilde{\beta}_{2})>d_{1}(\tilde{\beta}_{1})$. To prove
$c_{1}(\tilde{\beta}_{1})>c_{1}(\tilde{\beta}_{2})$, we use the inequalities
$\min_{y\in[c_{0},x]}G(\tilde{\beta}_{1},y)\geq\min_{y\in[c_{0},x]}G(\tilde{\beta}_{2},y)\geq
d_{1}(\tilde{\beta}_{2})>d_{1}(\tilde{\beta}_{1}).$ The rest of the proof can
be done inductively using the same approach.

####  6.1.8. Proof of Lemma 2.8

By Lemma 2.5 we need to consider only
$x\in(\hat{d}(\alpha_{*}),K_{1}-\delta)\cup(K_{3}+\delta,\hat{c}(\alpha_{*}))$.
When $x_{0}\in(\hat{d}(\alpha_{*}),K_{1}-\delta)$, we have
$x_{0}\in(d_{k_{1}}(\alpha_{*}),K_{1}-\delta)$ for some $k_{1}\in\mathbb{N}$.
In this case
$x_{0}<x_{1}=G(\alpha_{1},x_{0})<G(\alpha_{*},x_{0})<c_{k_{1}-1}(\alpha_{*})<\hat{c}(\alpha_{*})$,
and therefore, $x_{1}$ gets into either $(x_{0},K_{1}-\delta)$ or
$(K_{1}-\delta,K_{3}+\delta)$ or $(K_{3}+\delta,c_{k_{1}-1}(\alpha_{*}))$. In
the second case, a solution stays in $(K_{1}-\delta,K_{3}+\delta)$ forever,
see Lemma 2.5. In the last case $G(\alpha_{*},x_{1})>d_{k_{1}-1}(\alpha_{*})$
by Lemma 2.7(i).

So we are only interested in the situation when
$x_{1}\in(K_{3}+\delta,c_{k_{1}-1}(\alpha_{*}))$. We have then
$x_{1}>x_{2}=G(\alpha_{2},x_{1})>G(\alpha_{*},x_{1})>d_{k_{1}-1}(\alpha_{*})$,
and therefore, $x_{2}$ gets into either $(K_{3}+\delta,x_{1})$ or
$(K_{1}-\delta,K_{3}+\delta)$ or $(d_{k_{1}-1}(\alpha_{*}),K_{1})$.
Circulation can happen only in the last case if $x_{3}$ gets into
$(K_{3}+\delta,c_{k_{1}-2}(\alpha_{*}))$. Applying Lemma 2.7 (i), we conclude
that a solution cannot make transition between
$(K_{3}+\delta,\,c_{k_{1}}(\alpha_{*}))$ and
$(d_{k_{1}}(\alpha_{*}),K_{1}-\delta)$ more than $k_{1}+1$ times, and after
each transition, a solution moves one level closer to
$(K_{1}-\delta,K_{3}+\delta)$, in the sense that now possible transitions are
between $(K_{3}+\delta,\,c_{k_{1}-1}(\alpha_{*}))$ and
$(d_{k_{1}-1}(\alpha_{*}),K_{1}-\delta)$, i.e. if before the transition a
solution was in $(d_{i}(\alpha_{*}),\,c_{i}(\alpha_{*}))$, after the next
round of transitions a solution is in
$(d_{i-1}(\alpha_{*}),\,c_{i-1}(\alpha_{*}))$. Therefore it either enters
$(K_{1}-\delta,K_{3}+\delta)$ after a finite number of steps or remains in
$(x_{0},K_{1}-\delta)\cup(K_{3}+\delta,c_{k_{1}}(\alpha_{*}))$ and converges
to either $K_{1}$ or $K_{3}$.

Similar reasoning is applied for
$x_{0}\\!\in\\!(K_{3}+\delta,\hat{c}(\alpha_{*}))$.

####  6.1.9. Proof of Lemma 2.9

(i) If $\underline{\alpha}=\underline{\alpha}_{0}$ we have
$\max\limits_{x\in[K_{0},K_{1}]}G(\beta,x)<\hat{c}(\beta)$ or
$\min\limits_{y\in[K_{3},g_{m}]}G(\beta,y)>\hat{d}(\beta)$ for any
$\beta>\underline{\alpha}_{0}$ . Assume that
$\max\limits_{x\in[K_{0},K_{1}]}G(\beta,x)<\hat{c}(\beta)$. Since
$\hat{c}=\lim_{k\to\infty}c_{k}$, there exists a number $k_{0}$ s.t.
$\max\limits_{x\in[K_{0},K_{1}]}G(\beta,x)<c_{k_{0}}<\hat{c}(\beta)$. By
definition (2.6) this implies $d_{k+1}=d_{k}$ for $k\geq k_{0}+1$ and
$\hat{d}(\beta)=d_{k_{0}}$. The case
$\min\limits_{y\in[K_{3},g_{m}]}G(\beta,y)>\hat{d}(\beta)$ is similar. If
condition (2.8) holds, $\forall\beta\in(\beta_{0},1)$ we have
$k_{0}(\beta)<\infty$. Suppose $d_{k}(\beta)=d_{k_{0}}(\beta)$, for all $k\geq
k_{0}(\beta)$, then
$\max\limits_{x\in[K_{0},K_{1}]}G(\beta,x)<c_{k_{0}-1}(\beta)<\hat{c}(\beta)$,
which implies $\beta_{1}=\beta_{0}$. The case when
$c_{k}(\beta)=c_{k_{0}}(\beta)$ for all $k\geq k_{0}$ is similar.

(ii) If $\underline{\alpha}>\underline{\alpha}_{0}$, there exists $\beta$,
$\underline{\alpha}<\beta<\underline{\alpha}_{0}$ s.t.
$\max_{x\in[K_{0},K_{1}]}G(\beta,x)>\hat{c}(\beta)$ or
$\min_{x\in[K_{3},g_{m}]}G(\beta,x)>\hat{d}(\beta)$. Assume the first
inequality. Since $(c_{n}(\beta))_{n\in\mathbb{N}}$ is non-decreasing, we have
$\max_{x\in[K_{0},K_{1}]}G(\beta,x)>c_{k}(\beta)$ for each $k\in\mathbb{N}$,
so (2.6) implies that $(d_{n}(\beta))_{n\in\mathbb{N}}$ continues to decrease
and does not stop. So $(c_{n}(\beta))_{n\in\mathbb{N}}$ also continues to
increase and does not stop. Then condition (2.8) does not hold, which implies
that $\bar{\kappa}(\beta)>\hat{c}(\beta)>c_{k}(\beta)$ and
$\underline{\kappa}(\beta)<\hat{d}(\beta)<d_{k}(\beta)$ for all
$k\in\mathbb{N}$. So
$\min_{y\in[K_{3},g_{m}]}G(\beta,y)=G(\beta,\bar{\kappa}(\beta))<G(\beta,\hat{c}(\beta))=\lim_{k\to\infty}G(\beta,c_{k})=\lim_{k\to\infty}d_{k}=\hat{d}(\beta)$.
Analogously, $\max_{x\in[K_{0},K_{1}]}G(\beta,x)>\hat{c}(\beta)$. Therefore
$\beta<\underline{\alpha}$.

####  6.1.10. Proof of Theorem 2.10

Let $x_{0}\in(K_{0},K_{1}-\delta)$. Take some $\alpha_{*}>\underline{\alpha}$
and assume first that
$\displaystyle\max_{x\in[K_{0},K_{1}-\delta]}G(\alpha_{*},x)~{}<~{}\hat{c}(\alpha_{*})$.
Since $\hat{c}(\alpha_{*})$ is a limit of a non-decreasing sequence, there
exists $k_{1}$ s.t.
$\max_{x\in[K_{0},K_{1}-\delta]}G(\alpha_{*},x)<c_{k_{1}}(\alpha_{*})<\hat{c}(\alpha_{*})$.
From this place we just follow the scheme for the proof of Lemma 2.8. Similar
approach applies for $x_{0}\\!\in\\!(K_{3}+\delta,g_{m})$ and
$\inf_{x\in(K_{3}+\delta,g_{m})}G(\alpha_{*},x)>~{}\hat{d}(\alpha_{*})$, as
well as other cases.

####  6.1.11. Proof of Theorem 2.12

Fix some $\alpha$ and $\ell$ as in (2.12). Note that it is enough to consider
only $x_{0}\in(g^{2}_{m},g_{m})$. By (2.12) we have
$\alpha-\ell>\underline{\alpha}_{0},\,\alpha+\ell>\underline{\alpha}$,
$\alpha+\ell>\alpha_{n}>\alpha-\ell$, $n\in\mathbb{N}$. Fix some
$\displaystyle\varepsilon\in(0,1-(\underline{\alpha}-\underline{\alpha}_{0})/2)$
s.t. $\alpha+\ell>\underline{\alpha}+\varepsilon\ell$. Set

$\alpha_{*}:=\alpha+(1-\varepsilon)\ell,\quad\alpha^{*}:=\alpha+\ell.$

Since $\alpha_{*}>\underline{\alpha}$ and by (2.6), (2.10), there exists
$k_{0}=k_{0}(\alpha_{*})$ s.t.

$\sup_{x\in[K_{0},K_{1}-\delta]}\\!\\!\\!G(\alpha_{*},x)<c_{k_{0}}(\alpha_{*})\leq\hat{c}(\alpha_{*})\mbox{
or
}\inf_{y\in(K_{3}+\delta,g_{m}]}\\!\\!\\!G(\beta,y)>d_{k_{0}+1}(\alpha_{*})>\hat{d}(\alpha_{*}).$

Applying (2.6) and Lemma 2.7 (i)-(ii), we conclude that any solution $x$ to
(2.11) with $\alpha_{n}\equiv\alpha_{*}$ and
$x_{0}\in(K_{0},K_{1}-\delta]\cup[K_{3}+\delta,g_{m}]$ either remains in
$[g^{2}_{m},K_{1}-\delta]$ or $[K_{3}+\delta,g_{m}]$ or circulates between
these two intervals. If it remains in $[g^{2}_{m},K_{1}-\delta]$, it will
exceed $K_{1}-\delta$ in $N_{1}$ steps. Similarly, if the solution remains in
$[K_{3}+\delta,g_{m}]$, it will be less than $K_{3}+\delta$ in $N_{2}$ steps,
where

$N_{1}:=\left[\frac{K_{1}-\delta-g^{2}_{m}}{\min_{x\in[g^{2}_{m},K_{1}-\delta]}\\{G(\alpha_{*},x)-x\\}}\right]+1,$
$N_{2}:=\left[\frac{g_{m}-K_{3}-\delta}{\min_{x\in[K_{3}+\delta,g_{m}]}\\{x-G(\alpha_{*},x)\\}}\right]+1.$

The circulation between those intervals cannot be more than
$2k_{0}(\alpha_{*})$ times. Let $N:=2k_{0}(\alpha_{*})(N_{1}+N_{2})$, then in
$N$ steps the solution to (2.11) with $\alpha_{n}\equiv\alpha_{*}$ reaches
$[K_{1}-\delta,K_{3}+\delta]$.

By Lemma 2.11, there exists a random moment $\mathcal{N}$, s.t., with
probability 1, for $N$-steps in a row, starting from $\mathcal{N}$,

$\alpha_{n}=\alpha+\ell\xi_{n}>\alpha+(1-\varepsilon)\ell=\alpha_{*},\quad
n=\mathcal{N},\mathcal{N}+1,\dots,\mathcal{N}+N.$

Acting as in the proof of Lemma 2.8 and using $x_{\mathcal{N}(\omega)}$
instead of $x_{0}$ for each $\omega\in\Omega$, we conclude that

$x_{\mathcal{N}(\omega)+n(\omega)}\in[K_{1}-\delta,\,K_{3}+\delta]\quad\mbox{for
some integer}\quad n(\omega)\leq 2k_{0}(\alpha_{*}),\,\,\omega\in\Omega.$

Since $\alpha_{n}>\underline{\alpha}_{0}$ for each $n\in\mathbb{N}$ on all
$\Omega$, and by Lemma 2.5, as soon as the solution gets into
$[K_{1}-\delta,\,K_{3}+\delta]$, it stays there and converges either to
$K_{3}$ or to $K_{1}$, which concludes the proof.

###  6.2. Estimation of $\underline{\alpha}$

In this section we discuss the case mentioned in Remark 2. Below
$\underline{\alpha}_{0}$, $d_{i}(\beta)$, $c_{i}(\beta)$ ($\beta\in(0,1)$ and
$i\in\mathbb{N}$), $k_{0}(\beta)$, $\underline{\alpha}$ are defined as in
(2.2), (2.6), (2.7) and (2.10), respectively. Suppose that Assumption 1 and
condition (2.4) hold, $x_{m}$ is the only point of maximum of $g$ on
$[K_{0},K_{1}-\delta]$, $g$ decreases on $[K_{3},g_{m}]$, where
$g_{m}=g(x_{m})$, $g$ is differentiable outside of $(d_{1}(0),c_{1}(0))$ and,
for some $L_{1},L_{2}>0$, $g^{\prime}(x)>-L_{1}$ for $x\in[x_{m},d_{1}(0)]$
and $g^{\prime}(x)>-L_{3}$ for $x\in[c_{1}(0),g_{m}]$, where, for simplicity,
$L_{1}\geq L_{3}$. Note that, since $g$ decreases on $[x_{m},K_{1}]$, we have
$d_{1}(0)=g^{-1}(K_{3})$ and $c_{1}(0)=g^{-1}(d_{1}(0))$. Also, for any
$\alpha$ the function $G(\alpha,\cdot)$ has a maximum
$G_{\max}(\alpha)=G(\alpha,x_{Gm})$ on $(K_{0},K_{1})$ and
$x_{Gm}(\alpha)\in(x_{m},K_{1})$.

Define

$\underline{\alpha}_{1}:=\max\left\\{\underline{\alpha}_{0},\,\,\frac{L_{1}L_{3}-1}{(L_{1}+1)(L_{3}+1)}\right\\}.$

If we prove that $\underline{\alpha}\leq\underline{\alpha}_{1}$ then
$\underline{\alpha}_{1}$ can be used for stabilization even though it might
not coincide with the best (minimal) possible control.

By direct calculations we show the following.

  1. (i)

$\frac{L_{1}L_{3}-1}{(L_{1}+1)(L_{3}+1)}<\frac{L_{3}}{L_{3}+1}<\frac{L_{1}}{L_{1}+1}.$

  2. (ii)

If $\alpha<\frac{L_{i}}{L_{i}+1}$ then $(1-\alpha)L_{i}-\alpha>0$, $i=1,3$.

  3. (iii)

If $\alpha\in\left(\underline{\alpha}_{1},\frac{L_{3}}{L_{3}+1}\right)$ then
$\bigl{(}(1-\alpha)L_{3}-\alpha\bigr{)}\bigl{(}(1-\alpha)L_{1}-\alpha\bigr{)}<1$.
This holds since $z=\frac{L_{1}L_{3}-1}{(L_{1}+1)(L_{3}+1)}$ is the smallest
root of the equation
$\bigl{(}(1-z)L_{3}-z\bigr{)}\bigl{(}(1-z)L_{1}-z\bigr{)}=1$.

By Lemma 3.1, for each $k\in\mathbb{N}$, we have $\hat{d}(0)\leq
d_{k}(0)<K_{1}-\delta$, $\hat{c}(0)\geq c_{k}(0)>K_{1}+\delta$,
$g(\hat{d}(0))=\hat{c}(0)$, $g(\hat{c}(0))=\hat{d}(0)$, and, for each
$\alpha\in(0,1)$, $G(\alpha,\hat{d}(0))<g(\hat{d}(0))=\hat{c}(0)$,
$G(\alpha,\hat{c}(0))>g(\hat{c}(0))=\hat{d}(0)$, $d_{k}(0)>d_{k}(\alpha)$,
$c_{k}(0)<c_{k}(\alpha)$. Also, $\hat{d}(0)>\hat{d}(\alpha)$ and
$\hat{c}(0)<\hat{c}(\alpha)$. To prove that the last two inequalities are
strict, we show that $\\{\hat{d}(0),\bar{c}(0)\\}$ cannot be a two-cycle for
$G(\alpha,\cdot)$ with some $\bar{c}(0)\geq\hat{c}(0)$ and each
$\alpha\in(0,1)$. Indeed, assuming the contrary, we get

$\bar{c}(0)=G(\alpha,\hat{d}(0))=(1-\alpha)g(\hat{d}(0))+\alpha\hat{d}(0)=(1-\alpha)\hat{c}(0)+\alpha\hat{d}(0)\geq\hat{c}(0),$

which leads to a contradiction $\hat{c}(0)\leq\hat{d}(0)$.

Since there is no desired stability for the original function $g$, i.e.
$k_{0}(0)=\infty$, we have $x_{m}<\hat{d}(0)<d_{k}(0)$. Also, if
$k_{0}(\alpha)=\infty$ we have
$x_{m}<x_{Gm}(\alpha)<\hat{d}(\alpha)<d_{k}(\alpha)$.

Suppose first that
$\underline{\alpha}_{0}<\frac{L_{1}L_{3}-1}{(L_{1}+1)(L_{3}+1)}$. Let
$\alpha_{*}\in\left(\underline{\alpha}_{1},\frac{L_{3}}{L_{3}+1}\right)$ and
assume that $k_{0}(\alpha_{*})=\infty$. Then
$x_{m}<x_{Gm}(\alpha_{*})<\hat{d}(\alpha_{*})<d_{k}(\alpha_{*})<d_{1}(0)$,
$c_{1}(0)<\hat{c}(0)$. If we show that
$\\{\hat{d}(\alpha_{*}),\hat{c}(\alpha_{*})\\}$ is not a two-cycle for
$G(\alpha_{*},\cdot)$, it would contradict to the assumption
$k_{0}(\alpha_{*})=\infty$ and prove that
$\underline{\alpha}>\underline{\alpha}_{1}$.

Let $x\in[x_{m},d_{1}(0)]$ be such that $G(\alpha_{*},x)\in[c_{1}(0),g_{m}]$,
then

$\begin{split}&G(\alpha_{*},x)-\hat{c}(0)=G(\alpha_{*},x)-g(\hat{d}(0))<G(\alpha_{*},x)-G(\alpha_{*},\hat{d}(0))\\\
=&(1-\alpha_{*})[g(x)-g(\hat{d}(0))]+\alpha_{*}(x-\hat{d}(0))\leq[(1-\alpha_{*})L_{1}-\alpha_{*}](\hat{d}(0)-x).\end{split}$
(6.2)

Since $c_{1}(0)<\hat{c}(0)<g_{m}$,

$\displaystyle\hat{d}(0)-G^{2}(\alpha_{*},x)=$ $\displaystyle
g(\hat{c}(0))-G^{2}(\alpha_{*},x)\leq
G(\alpha_{*},\hat{c}(0))-G^{2}(\alpha_{*},x)$ $\displaystyle=$
$\displaystyle(1-\alpha_{*})[g(\hat{c}(0))-g(G(\alpha_{*},x))]+\alpha_{*}(\hat{c}(0)-G(\alpha_{*},x))$
$\displaystyle\leq$
$\displaystyle[(1-\alpha_{*})L_{3}-\alpha_{*}](G(\alpha_{*},x)-\hat{c}(0)).$

Since $\alpha_{*}<\frac{L_{3}}{L_{3}+1}$, by (ii)-(iii) we have
$(1-\alpha_{*})L_{i}-\alpha_{*}>0$, $i=1,2$,
$\prod_{i=1}^{2}[(1-\alpha_{*})L_{i}-\alpha_{*}]<1$ and therefore

$\displaystyle\hat{d}(0)-G^{2}(\alpha_{*},x)\leq[(1-\alpha_{*})L_{3}-\alpha_{*}][(1-\alpha_{*})L_{1}-\alpha_{*}](\hat{d}(0)-x)<\hat{d}(0)-x,$

which implies $G^{2}(\alpha_{*},x)>x$. If
$\\{\hat{d}(\alpha_{*}),\hat{c}(\alpha_{*})\\}$ is a two-cycle for
$G(\alpha_{*},\cdot)$ then
$G(\alpha_{*},\hat{d}(\alpha_{*}))=\hat{c}(\alpha_{*})\in[c_{1}(0),g_{m}]$,
$G(\alpha_{*},\hat{c}(\alpha_{*}))=\hat{d}(\alpha_{*})\in[x_{m},d_{1}(0)]$ and

$G^{2}(\alpha_{*},\hat{d}(\alpha_{*}))=G(\alpha_{*},G(\alpha_{*},\hat{d}(\alpha_{*})))=G(\alpha_{*},\hat{c}(\alpha_{*}))=\hat{d}(\alpha_{*}).$

However, from the above, we should have
$G^{2}(\alpha_{*},\hat{d}(\alpha_{*}))>\hat{d}(\alpha_{*})$, which is a
contradiction.

If $\alpha_{*}>\frac{L_{1}}{L_{1}+1}$ then $(1-\alpha_{*})L_{1}-\alpha_{*}<0$
and from (6.2) we conclude that $G(\alpha_{*},x)<\hat{c}(0)$, which implies
that for some $k\in\mathbb{N}$, we have
$G(\alpha_{*},x)<c_{k}(0)<c_{k}(\alpha_{*})\leq\hat{c}(\alpha_{*})$, so
$\alpha_{*}>\underline{\alpha}$, see (2.10). If
$G(\alpha_{*},x)\in[\hat{c}(0),g_{m}]$ and
$\frac{L_{1}}{L_{1}+1}>\alpha_{*}>\frac{L_{3}}{L_{3}+1}$, we get
$G^{2}(\alpha_{*},x)>\hat{d}(0)\geq x$. From here we proceed as above and get
a contradiction for $x=\hat{d}(\alpha_{*})$.

Assume now that
$\underline{\alpha}_{0}\geq\frac{L_{1}L_{3}-1}{(L_{1}+1)(L_{3}+1)}$. The cases
$\alpha_{*}>\frac{L_{1}}{L_{1}+1}$ and
$\frac{L_{1}}{L_{1}+1}>\alpha_{*}>\frac{L_{3}}{L_{3}+1}$ were considered
above. If $\frac{L_{3}}{L_{3}+1}>\underline{\alpha}_{0}$ then the case
$\frac{L_{3}}{L_{3}+1}>\alpha_{*}>\underline{\alpha}_{0}$ has already been
discussed.

###  6.3. Control for an arbitrary number of equilibrium points

####  6.3.1. Classification of intervals

We set, for each $i\in I^{+}$ and $i_{0}$ defined in (3.1),

$l_{+-}(i):=\left\\{\begin{array}[]{ll}\inf\\{j>i:j\in
I^{-}\\},&\mbox{if}\quad\\{j>i:j\in I^{-}\\}\neq\emptyset,\\\ i_{0},&\mbox{\rm
otherwise},\end{array}\right.$ (6.3)

and for each $i\in I^{-}$

$l_{-+}(i):=\left\\{\begin{array}[]{ll}\inf\\{j>i:j\in
I^{+}\\},&\mbox{if}\quad\\{j>i:j\in I^{+}\\}\neq\emptyset,\\\ i_{0},&\mbox{\rm
otherwise}.\end{array}\right.$ (6.4)

Introduce the set of all consecutive intervals
${\mathcal{V}}:=\\{(K_{p},K_{p+1}),K_{p},K_{p+1}\in\mathbb{K},\,p=0,1,\dots,j_{0}-1\\}$
with the ends in the set of equilibrium points $\mathbb{K}$, see (3.2). We
create two stages of control: with $\alpha^{(1)}$ and $\alpha^{(2)}$,
$1>\alpha^{(2)}\geq\alpha^{(1)}>\bar{L}/(\bar{L}+1)$. We also distinguish
between two types of blocks of consecutive intervals from $\mathcal{V}$. After
application of the first stage of control,
$\alpha_{n}\in\left(\alpha^{(1)},\,1\right)$, there will be no communication
between any blocks. After application of the second stage of control,
$\alpha_{n}\in\left(\alpha^{(2)},1\right)$, there will be no infinite
circulation inside of any block.

There are two blocks of the first type, $\mathcal{V}_{0}$ and
$\tilde{\mathcal{V}}$. $\mathcal{V}_{0}$ consists of the initial intervals
from $\mathcal{V}$ with indexes $p$ from $0$ to some $m_{0}$,
$\tilde{\mathcal{V}}$ is made of the intervals with indexes $p$ from some
$\tilde{m}$ to $j_{0}-1$. After the application of the first control stage, if
a solution gets into either $\mathcal{V}_{0}$ or $\tilde{\mathcal{V}}$, it
stays in the block and converges to one of equilibrium points. There are
$s_{0}$ blocks in the second group, and each block of the second type contains
two groups of intervals, between which circulation is possible. In other
words,

${\mathcal{V}}=\mathcal{V}_{0}\bigcup\left[\bigcup_{s=0}^{s_{0}-1}\mathcal{V}_{2s+1}\right]\bigcup\tilde{\mathcal{V}}.$
(6.5)

Note that $\mathcal{V}_{0}$ or $\tilde{\mathcal{V}}$, or all of
$\mathcal{V}_{2s+1}$ can be empty.

Now we proceed to a detailed definition of blocks. Assume $0\in I^{-}$, i.e.
the left Lipschitz constant at $K_{1}$ is finite and less than the right
Lipschitz constant at $K_{1}$, see (3.5) and (3.6). Denote

$\mathcal{V}_{0}:=\\{(K_{p},K_{p+1}),0\leq p\leq
2m_{0}-1\\},~{}~{}m_{0}:=l_{-+}(0)\in I^{+},$

see (6.4), so $K_{2m_{0}+1}$ is the first from the left equilibrium where the
right Lipschitz constant is finite and less than the left Lipschitz constant.

By Lemma 3.1, a solution remains in any interval
$[K_{2j},K_{2j+1}]\in\mathcal{V}_{0}$ forever. If $x_{0}\in[K_{2j-1},K_{2j}]$,
$j\leq m_{0}$, then a solution can get out of that interval but only to the
left and then remains in one of $[K_{2i},K_{2i+1}]$, $0\leq i\leq j-1$. So
circulation is impossible if $x_{0}$ is from one of the intervals of
$\mathcal{V}_{0}$. Note that if $0\in I^{+}$, the set $\mathcal{V}_{0}$ is
empty and $m_{0}=0$.

Denote now $\mathcal{K}_{0,1}:=\left\\{[K_{p},\,K_{p+1}],\,2m_{0}\leq p\leq
2m_{1}-1\right\\}$, where $m_{1}=l_{+-}(m_{0})\in I^{-}$, see (6.4), so
$K_{2m_{1}+1}$ is the first equilibrium after $K_{2m_{0}}$ where the left
Lipschitz constant is finite and less than the right Lipschitz constant. From
any interval $(K_{2j},\,K_{2j+1})\in\mathcal{K}_{0,1}$, a solution can move to
the right. If it gets into $[K_{2j+1},\,K_{2j+2}]\in\mathcal{K}_{0,1}$, it
stays there. The first case of circulation is possible when a solution jumps
over $K_{2m_{1}}$ to the interval of type $[K_{2j+1},\,K_{2j+2}]$ from
$\mathcal{K}_{1,2}:=\left\\{(K_{p},\,K_{p+1}),\,2m_{1}\leq p\leq
2m_{2}-1\right\\}$, $m_{2}:=l_{-+}(m_{1})\in I^{+}$. Infinite circulation
happens when a solution attends infinitely many times some intervals of type
$(K_{2j},\,K_{2j+1})$, in ascending order, from $\mathcal{K}_{0,1}$, and some
intervals of type $(K_{2k+1},\,K_{2k+2})$, in descending order, from
$\mathcal{K}_{1,2}$, where $j\leq m_{1}-1$ and $k\geq m_{1}$, since, by Lemma
3.1, if a solution gets into the interval $(K_{2m_{1}-1},\,K_{2m_{1}+1})$, it
stays in it.

We denote
$\mathcal{V}_{1}:=\mathcal{K}_{0,1}\cup\mathcal{K}_{1,2}=\left\\{[K_{p},\,K_{p+1}],\,2m_{0}\leq
p\leq 2m_{2}-1\right\\}$. Inductively, we define
$\mathcal{K}_{2s,2s+1}:=\left\\{(K_{p},\,K_{p+1}),\,2m_{2s}\leq p\leq
2m_{2s+1}-1\right\\}$ with $m_{2s+1}:=l_{+-}(m_{2s})\in I^{-}$, and
$\mathcal{K}_{2s+1,2s+2}:=\left\\{(K_{p},\,K_{p+1}),\,2m_{2s+1}\leq p\leq
2m_{2s+2}-1\right\\}$ with $m_{2s+2}:=l_{-+}(m_{2s+1})\in I^{+}$,

$\begin{array}[]{l}\displaystyle\mathcal{V}_{2s+1}:=\mathcal{K}_{2s,2s+1}\cup\mathcal{K}_{2s+1,2s+2}=\left\\{[K_{p},\,K_{p+1}],\,2m_{2s}\leq
p\leq 2m_{2s+2}-1\right\\},\\\ s\leq s_{0}-1,\end{array}$ (6.6)

where $s_{0}=\max\\{s:2m_{2s}\leq j_{0}\\}$. Similarly, infinite circulation
happens when a solution attends infinitely many times some intervals of type
$(K_{2j},\,K_{2j+1})$, in ascending order, from $\mathcal{K}_{2s,2s+1}$, and
some intervals of type $(K_{2k+1},\,K_{2k+2})$, in descending order, from
$\mathcal{K}_{2s+1,2s+2}$, where $j\leq m_{s}-1$ and $k\geq m_{s}$, since, by
Lemma 3.1, if a solution gets into the interval
$(K_{2m_{s}-1},\,K_{2m_{s}+1})$ it stays in it.

To illustrate this, we use again the fourth iterate of Ricker’s map, see Fig.
2 and Section 3, and have $m_{j}=j$, $j=0,\dots,4$,
$\mathcal{K}_{0,1}=\\{(K_{0},K_{1}),\,(K_{1},K_{2})\\}$,
$\mathcal{K}_{1,2}=\\{(K_{2},K_{3}),\,(K_{3},K_{4})\\}$,
$\mathcal{K}_{2,3}=\\{(K_{4},K_{5}),\,(K_{5},K_{6})\\}$,
$\mathcal{K}_{3,4}=\\{(K_{6},K_{7}),\,(K_{7},\infty)\\}$.

If $2m_{2s_{0}}=j_{0}$ then $\tilde{\mathcal{V}}=\emptyset$. If
$2m_{2s_{0}}<j_{0}$ then there is $i^{*}$ s.t. either
$2m_{2s_{0}+i^{*}}=j_{0}$ or $2m_{2s_{0}+i^{*}}+1=j_{0}$. In both cases we
have $m_{2s_{0}+i}\in I^{+}$ for all $i=0,1,\dots,i^{*}$, see (3.6), and we
can define non-empty

$\tilde{\mathcal{V}}:=\left\\{[K_{p},\,K_{p+1}],\,2m_{2s_{0}}\leq p\leq
j_{0}-1\right\\}.$

Note that $j\in I^{+}$ in each interval $(K_{2j},K_{2j+1})$ and
$(K_{2j+1},K_{2j+2})$ from $\tilde{\mathcal{V}}$. Thus, from each
$[K_{2j},K_{2j+1}]\in\tilde{\mathcal{V}}$, a solution can move only to the
right and not further than $K_{j_{0}}$, and it remains in each
$[K_{2j+1},K_{2j+2}]\in\tilde{\mathcal{V}}$.

All the above gives us the decomposition in (6.5). Therefore, for each
function $g$ satisfying Assumption 3, the interval $[K_{0},K_{j_{0}}]$ can be
represented as a union of blocks (6.5). The only blocks where a cycle can
occur are blocks $\mathcal{V}_{2s+1}$ with $0\leq s\leq s_{0}-1$.

In Sections 6.3.2-6.4.1, for a block of this type we briefly discuss how to
introduce a new control $\beta_{1}\in\left(\bar{L}/(\bar{L}+1),\,1\right)$,
which eliminates the circulation.

####  6.3.2. Deterministic control

Consider one of the blocks $\mathcal{V}_{2s+1}$ defined by (6.6) and
consisting of two adjacent groups $\mathcal{K}_{2s,2s+1}$ and
$\mathcal{K}_{2s+1,2s+2}$ with the lengths of $m$ and $r$, respectively. To
simplify the description of the structure for $\mathcal{V}_{2s+1}$, we shift
indexes to zero, i.e. $s=0$. We assume that the block ${\mathcal{V}}_{1}$
contains equilibrium points

$\begin{array}[]{l}\displaystyle\mathbb{K}_{1}=\left\\{K_{0},K_{1},\dots
K_{2m-1},K_{2m},K_{2m+1},K_{2m+2},\dots,K_{2(m+r)}\right\\}\in\mathbb{K},\\\
m\in\mathbb{N}_{0},\,r\in\mathbb{N},\,r\geq 2\end{array}$ (6.7)

and satisfies

######  Assumption 5.

In the set of equilibrium points defined by (6.7), $0,1,2,\dots,m\in I^{+}$
and also $m+1,\dots m+r\in I^{-}$ if $K_{2(m+r)}<\infty$, while $m+1,\dots
m+r-1\in I^{-}$ if $K_{2(m+r)}=\infty$, where $I^{+}$ and $I^{-}$ are defined
as in (3.6).

Analogously to Theorem 2.12, we can formulate and prove the result about
stability on the block $\mathcal{V}_{1}$, see Lemma 6.1.

Assumption 5 implies that $m+1=l_{+-}(0)\in I^{-}$, i.e. the left Lipschitz
constants at all $K_{2j+1}$, $j\leq m$, are finite and less than the right
Lipschitz constants, see definitions (6.4), and for $j>m$ the opposite is
valid, so $\mathcal{K}_{0,1}=\\{(K_{p},K_{p+1}),\,\,0\leq p\leq 2m\\}$,
$\mathcal{K}_{1,2}=\\{(K_{p},K_{p+1}),\,\,2m\leq p\leq 2(m+r)-1\\}$. We define

$\beta_{0}:=\inf\left\\{\beta\in\left(\frac{\bar{L}}{\bar{L}+1},1\right)\\!:\\!\\!\\!\\!\inf_{x\in(K_{1},K_{2(m+r)})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!G(\beta,x)>K_{0},\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sup_{x\in\left(K_{0},K_{2(m+r)-1}\right)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!G(\beta,x)<K_{2(m+r)}\right\\},$
(6.8)

where for $K_{2(m+r)}=\infty$ the second inequality in the right-hand side of
(6.8) holds unconditionally.

Fix some $\beta\in\left(\beta_{0},\,1\right)$. Definition (6.8) guarantees

$G(\beta,\cdot):(K_{0},K_{2(m+r)})\to(K_{0},K_{2(m+r)}).$

Also, applying Lemma 3.1 we conclude that

$G(\beta,\cdot):(K_{2m+1},K_{2m+3})\to(K_{2m+1},K_{2m+3})$

and that any cycle is possible only outside of $(K_{2m+1},K_{2m+3})$.
Following the procedure introduced in Section 2.3, we extend the interval
$(K_{2m+1},K_{2m+3})$ to keep this property and then introduce the smallest
$\beta$, for which infinite circulation of a solution between
$\mathcal{K}_{0,1}$ and $\mathcal{K}_{1,2}$ becomes impossible.

Reasoning as in Section 2.2, we assume that
$\max_{y\in[K_{0},K_{2m+1}]}g(x)=:g_{m}>K_{2m+3}$ and consider only such
$\beta$ that $\max_{y\in[K_{0},K_{2m+1}]}G(\beta,x)>K_{2m+3}$. Define now
$\underline{\kappa}(\beta)$, the largest point of maximum of $G(\beta,\cdot)$
on $(K_{0},\,K_{2m+1})$, and $\bar{\kappa}(\beta)$, the smallest point of
minimum of $G(\beta,\cdot)$ on $(K_{2m+3},\,g_{m})$:
$\underline{\kappa}(\beta)=\sup\\{y\in\left(K_{0},\,K_{2m+1}\right):G(\beta,y)=\sup_{x\in(K_{0},K_{2m+1})}G(\beta,x)\\},$
$\bar{\kappa}(\beta)=\inf\\{y\in\left(K_{2m+3},\,g_{m}\right):G(\beta,y)=\inf_{x\in(K_{2m+3},g_{m})}G(\beta,x)\\}$.
By (2.6) we introduce two convergent sequences of points,
$(d_{n}(\beta))_{n\in\mathbb{N}}$ and $(c_{n}(\beta))_{n\in\mathbb{N}}$,
located in $(K_{0},K_{2m+1})$ and in $(K_{2m+3},\,g_{m})$, respectively, only
with $d_{0}:=K_{2m+1}$, $c_{0}:=K_{2m+3}$. Now, as in Section 2.3, we define
$k_{0}(\beta)$, $\hat{c}(\beta)$, $\hat{d}(\beta)$ by (2.7), (2.9). Analogues
of Lemmata 2.7 and 2.8 hold when Assumption 1 is substituted by Assumptions
3,4,5. The interval $(\hat{d}(\beta),\hat{c}(\beta))$ includes
$(K_{2m+1},K_{2m+3})$ and is invariant under $G(\beta,\cdot)$. Note that
$k_{0}(\beta)=\infty$ if and only if $\\{\hat{c}(\beta),\hat{d}(\beta)\\}$ is
a two-cycle for $G(\beta,\cdot)$, so for this particular $\beta$ the interval
of the initial values with the desired convergence cannot be increased.
Moreover, the bound for control $\alpha$ is sharp: if it is smaller, a cycle
rather than an equilibrium can be an attractor. We introduce $\beta_{1}$ as

$\beta_{1}:=\inf\left\\{\beta\in(\beta_{0},1):\\!\\!\\!\\!\\!\\!\max_{x\in[K_{0},K_{2m+1}]}\\!\\!\\!\\!\\!\\!G(\beta,x)<\hat{c}(\beta)~{}\mbox{or}\\!\\!\\!\\!\inf_{y\in(K_{2m+3},g_{m})}\\!\\!\\!\\!\\!\\!G(\beta,y)>\hat{d}(\beta)\right\\},$
(6.9)

which is larger than $\beta_{0}$, and show that $\beta_{1}$ is well-defined.
The proof of the following lemma, which is the main result of this section
concerning stability on the block $\mathcal{V}_{1}$ is similar to the proof of
Theorem 2.10.

######  Lemma 6.1.

Let Assumptions 3, 4 and 5 hold, and $I$, $\beta_{1}$ be defined as in (3.1)
and (6.9). Let $\alpha_{*}\in(\beta_{1},1)$, $\alpha^{*}\in(\alpha_{*},1)$,
$\alpha_{n}\in[\alpha_{*},\alpha^{*}]$ for all $n\in\mathbb{N}$, and $x$ be a
solution to (1.2) with $x_{0}\in\left(K_{0},K_{2(m+r)}\right)$. Then $x$
converges either to some $K_{2i+1}$, $i\in I$, or to an equilibrium inside
$(K_{p},K_{p+1})$, $0\leq p\leq j_{0}-1$.

######  Remark 5.

Reasoning as in Lemmata 2.7 and 2.9, we can show that the choice of lower
bound $\beta_{1}$ by (6.9) is quite sharp: when $\beta_{1}>\beta_{0}$,
existence of a two-cycle for some $\beta>\beta_{0}$ confirms that not all
solutions to (1.2) converge to an equilibrium.

###  6.4. Proof of Theorem 3.2

According to Section 6.3.1, all the intervals $(K_{0},K_{j_{0}})$ can be
represented by (6.5) as a union of blocks of the intervals

$\mathcal{V}_{0}\bigcup\left[\bigcup_{s=0}^{s_{0}-1}\mathcal{V}_{2s+1}\right]\bigcup\tilde{\mathcal{V}},$

where inside of blocks $\mathcal{V}_{0}$ and $\tilde{\mathcal{V}}$ a solution
cannot circulate. The first stage of control, $\alpha^{(1)}\in(0,1)$, stops
communication between blocks. By Assumptions 3-4 and the form of the function
$G(\alpha,x)$, see (2.1), such $\alpha^{(1)}$ exists. Applying Lemma 6.1, for
each block of type $\mathcal{V}_{2s+1}$ we find a control $\beta_{1}(s)$ which
stops circulation inside of $\mathcal{V}_{2s+1}$. By setting
$\alpha_{*}>\max\\{\alpha^{(1)},\beta_{1}(s),s=0,\dots,s_{0}-1\\}$ we conclude
the proof.

####  6.4.1. Stochastic control

In this section we briefly discuss some situations when a control perturbed by
an additive noise, $\alpha+\ell\xi_{n}$, can improve the deterministic result,
decreasing the lower bound of $\alpha$.

######  Assumption 6.

Let in Assumptions 3,4,5 the set $\mathbb{K}$ be defined by (6.7), and
$g(x)\neq x$ for $x\in(K_{p},K_{p+1})$ and all $K_{p},K_{p+1}\in\mathbb{K}$.

Let $\beta_{0}$, $\underline{\kappa}(\beta)$, $d_{k}$, $\bar{\kappa}(\beta)$,
$c_{k}$ be as in Section 6.3.2 (see also Section 2.3). Circulation of a
solution is possible only if both
$G(\beta_{0},\underline{\kappa}(\beta_{0}))>K_{2m+3}$ and
$G(\beta_{0},\bar{\kappa}(\beta_{0}))<K_{2m+1}$ hold. In this case we can
define

$\begin{split}\tau:=\inf\\{s\in\\{0,\dots,m\\}:K_{2s+1}\geq\underline{\kappa}(\beta_{0})\\},\,\,&\mbox{where}\,\,\underline{\kappa}(\beta_{0})\in(K_{2\tau},K_{2\tau+1}),\\\
\theta:=\sup\\{s>m+1:K_{2s+1}\leq\bar{\kappa}(\beta_{0})\\},\,\,&\mbox{where}\,\,\bar{\kappa}(\beta_{0})\in(K_{2\theta+1},K_{2\theta+2}).\end{split}$
(6.10)

Notation (6.10) yields that $G(\beta_{0},\cdot)$ takes its maximum with
respect to all intervals to the left of $K_{2\tau}$ on
$[K_{2\tau},K_{2\tau+1}]$. Applying Lemma 2.2(iv), we can show that the same
holds for $G(\beta,\cdot)$, $\beta\in(\beta_{0},1)$. Similarly,
$G(\beta_{0},\cdot)$ takes its minimum with respect to all the intervals to
the right of $K_{2\theta+2}$ on $[K_{2\theta+1},K_{2\theta+2}]$, and the same
holds for $G(\beta,\cdot)$, $\beta\in(\beta_{0},1)$. Define now

$\begin{split}\beta_{21}=\inf&\left\\{\beta\in(\beta_{0},1):\\!\\!\\!\\!\sup_{x\in[K_{2\tau+1},K_{2m+1}]}\\!\\!\\!\\!\\!\\!G(\beta,x)<K_{2m+3},\right.\\\
&\left.\sup_{x\in[K_{2\tau},K_{2\tau+1}]}\\!\\!\\!\\!\\!\\!G(\beta,x)<K_{2\theta+2}\right\\},\\\
\beta_{22}=\sup&\left\\{\beta\in(\beta_{21},1):\sup_{x\in[K_{2\tau},K_{2\tau+1}]}G(\beta,x)>K_{2\theta+1}\right\\},\\\
\beta_{23}=\inf&\left\\{\beta\in(\beta_{0},1):\\!\\!\\!\\!\inf_{x\in[K_{2m+3},\,K_{2\theta+1}]}\\!\\!\\!\\!G(\beta,x)>K_{2\tau+1},\right.\\\
&\left.\inf_{x\in[K_{2\theta+1},\,K_{2\theta+2}]}\\!\\!\\!\\!\\!\\!G(\beta,x)>K_{2\tau}\right\\},\\\
\beta_{24}=\sup&\left\\{\beta\in(\beta_{23},1):\inf_{x\in[K_{2\theta+1},\,K_{2\theta+2}]}G(\beta,x)<K_{2\tau+1}\right\\}.\end{split}$
(6.11)

######  Remark 6.

If the set on the third line of (6.11) is empty then $\beta_{21}$ is the
stabilization lower bound. If the set on the last line is empty, we get the
stabilization lower bound $\beta_{23}$. Since in these two cases a
stabilization bound is known, we proceed to establishing the required $\beta$
when these sets are non-empty.

For $\beta\in(\beta_{21},\beta_{22})$ we have
$\sup_{x\in[K_{2\tau},K_{2\tau+1}]}G(\beta,x)\in(K_{2\theta+1},K_{2\theta+2})$,
and for $\beta\in(\beta_{23},\beta_{24})$ we have
$\inf_{x\in[K_{2\theta+1},\,K_{2\theta+2}]}G(\beta,x)\in(K_{2\tau},K_{2\tau+1})$.

If $\beta_{22}\leq\beta_{23}$ then
$\sup_{x\in[K_{2\tau},K_{2\tau+1}]}G(\beta,x)\leq K_{2\theta+1}$ for any
$\beta>\beta_{22}$, which means that the circulation of a solution between the
intervals $(K_{2\tau},K_{2\tau+1})$ and $(K_{2\theta+1},\,K_{2\theta+2})$ is
impossible. Therefore we consider only the case when $\beta_{22}>\beta_{23}$.

######  Assumption 7.

Assume that $\beta_{2j}$, $j=1,\dots,4$, be well defined by (6.11) and

$(\beta_{21},\beta_{22})\cap(\beta_{23},\beta_{24})=({\underline{\beta}}_{2},\bar{\beta}_{2})\not=\emptyset.$
(6.12)

######  Remark 7.

For $\beta\in({\underline{\beta}}_{2},\bar{\beta}_{2})$, Assumption 7 implies

$\begin{split}&\underline{\kappa}(\beta),d_{k}(\beta),\hat{d}(\beta)\in(K_{2\tau},K_{2\tau+1})\mbox{~{}~{}~{}and~{}}\\\
&\bar{\kappa}(\beta),c_{k}(\beta),\hat{c}(\beta)\in(K_{2\theta+1},K_{2\theta+2}),~{}k\in\mathbb{N}.\end{split}$
(6.13)

######  Remark 8.

Note that Assumption 7 means that there is a possibility of circulation of a
solution between intervals $(K_{2\tau},K_{2\tau+1})$ and
$(K_{2\theta+1},K_{2\theta+2})$. By the choice of $\beta_{ij}$ in (6.11), we
eliminate circulation inside of $(K_{2\tau+1},\,K_{2\theta+1})$, turning
$(K_{2\tau+1},K_{2\theta+1})$ into a new trap. By (6.11) and (6.12) the
solution cannot get above $K_{2\theta+2}$ from $(K_{0},K_{2\tau})$, and it
cannot get below $K_{2\tau}$ from $(K_{2\theta+1},K_{2(m+r)})$.

######  Remark 9.

If $\max_{x\in[K_{2m},K_{2m+1}]}g(x)\leq K_{2m+4}$ and
$\min_{x\in[K_{2m+3},K_{2m+4}]}g(x)\geq K_{2m}$ then
$\beta_{21}=\beta_{23}=\beta_{0}$, and if
$\min\\{\beta_{22},\beta_{24}\\}>\beta_{0}$, circulation between intervals is
possible. Such situation is illustrated in Section 4, Examples 1 and 3, where
$m=0$ and $r=2$.

To show that under Assumptions 6 and 7 introduction of the noise into control
can improve the deterministic result, we set

$\begin{split}\beta_{3}:=\inf&\left\\{\beta\in({\underline{\beta}}_{2},\bar{\beta}_{2}):\\!\\!\\!\\!\sup_{x\in(K_{2\tau},K_{2\tau+1})}\\!\\!\\!\\!G(\beta,x)<\hat{c}(\beta)\,\mbox{~{}~{}~{}or}\right.\\\
&\left.\inf_{y\in(K_{2\theta+1},K_{2\theta+2})}\\!\\!\\!\\!G(\beta,y)>\hat{d}(\beta)\right\\}.\end{split}$
(6.14)

Relations (6.11), (6.12) and (6.13) imply that $\beta_{3}$ is well defined.

######  Remark 10.

If $\underline{\beta}_{2}=\beta_{3}$ and
$\alpha\in(\underline{\beta}_{2},\bar{\beta}_{2})$, by Lemma 6.1, we do not
need to introduce a noise perturbation $\ell\xi_{n}$ to achieve stabilization
of all equilibria in $\mathbb{K}$. Any small noise perturbation with
$\ell<\min\\{\bar{\beta}_{2}-\alpha,\,\alpha-\underline{\beta}_{2}\\}$ keeps
this stability which was achieved by the deterministic part $\alpha$ of the
control $\alpha_{n}=\alpha+\ell\xi_{n}$. Theorem 6.2 below is devoted to the
case when $\underline{\beta}_{2}<\beta_{3}$, and the noise plays an active
role in stabilization of the equilibrium points $K_{2\tau+1}$ and
$K_{2\theta+1}$.

Choose some

$\alpha\in\left((\underline{\beta}_{2}+\beta_{3})/2,\,\bar{\beta}_{2}\right),\quad\ell\in\left(\beta_{3}-\alpha,\,\min\\{\alpha-\underline{\beta}_{2},\,\bar{\beta}_{2}-\alpha\\}\right).$
(6.15)

By the definition of $\beta_{3}$ in (6.14) and the choice of $\alpha$ in
(6.15), the second interval in (6.15) is non-empty.

######  Theorem 6.2.

Let Assumptions 3-7 hold. Let $\mathbb{K}$ be defined as in (6.7) and
$\alpha$, $\ell$ satisfy (6.15). Then any solution to (2.11) with
$x_{0}\in(K_{0},K_{2(m+r)})$ converges to one of the equilibrium points in
$\mathbb{K}$, with a total probability of one.

###### Proof.

Set $\tilde{G}(\beta_{0}):=\max_{x\in[K_{0},K_{2m+1}]}G(\beta_{0},x)$ for
$\beta_{0}$ as in (6.8). Choose $\alpha$ and $\ell$ satisfying (6.15)
arbitrarily. Since $\alpha+\ell\in\left(\beta_{3},\bar{\beta}_{2}\right)$, we
can find an $\varepsilon>0$ such that
$\alpha+(1-\varepsilon)\ell\in\left(\beta_{3},\,\bar{\beta}_{2}\right)$, and
also we have $\alpha-\ell>\underline{\beta}_{2}$. Relations (6.10), (6.11),
(6.12), (6.13) imply that whenever
$\beta\in({\underline{\beta}}_{2},\bar{\beta}_{2})$, we have
$\inf_{x\in(K_{2m+3},K_{2(m+r)})}G(\beta,x)>K_{2\tau},\quad\sup_{x\in(K_{0},K_{2m+1})}G(\beta,x)<\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}.$
Therefore there exists $\sigma=\sigma(\beta)>0$ s.t.

$\begin{split}&\inf_{x\in(K_{2m+3},K_{2(m+r)})}\\!\\!\\!\\!G(\beta,x)>K_{2\tau}+\sigma,\\\
&\sup_{x\in(K_{0},K_{2m+1})}\\!\\!G(\beta,x)<\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}-\sigma.\end{split}$
(6.16)

By Remark 6 we have
$\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}>K_{2\theta+1}$. If (6.16) holds
for a given $\beta_{1}$ and $\sigma$, it is also satisfied for the same
$\sigma$ and any $\beta\in(\beta_{1},1)$. So the inequality (6.16) holds for
any

$\sigma\leq\left\\{\sigma(\alpha-\ell),\,\,\frac{K_{2\tau+1}-K_{2\tau}}{2},\,\,\frac{\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}-K_{2\theta+1}}{2}\right\\}.$
(6.17)

Define, for $\beta\in[\alpha-\ell,\alpha+\ell]$,

$\delta(\beta):=\min\left\\{K_{2\tau+1}-d_{1}(\beta),\,\,c_{1}(\beta)-K_{2\theta+1}\right\\}.$
(6.18)

By (6.13) the right-hand side in (6.18) is positive, and, by Lemma 2.7 (vi),
we have $\delta(\alpha+\ell)\geq\delta(\beta)\geq\delta(\alpha-\ell)$. Define,
for $\sigma$ satisfying (6.17) and $\delta\leq\delta(\alpha-\ell)$, which is
small enough,

$\begin{split}&\Delta_{1}:=\\!\\!\min\left\\{G(\alpha+\ell,x)-x,\,x\in[K_{2\tau}+\sigma,K_{2\tau+1}-\delta]\right\\},\\\
&\Delta_{2}:=\\!\\!\min\\!\left\\{x-G(\alpha+\ell,x),\,x\in\left[K_{2\theta+1}+\delta,\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}-\sigma\right]\right\\},\\\
&N_{1}:=\left[\frac{K_{2\tau+1}-\delta-
K_{2\tau}-\sigma}{\Delta_{1}}\right]+1,\\\
&N_{2}:=\\!\\!\left[\frac{\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}-\sigma-
K_{2\theta+1}-\delta}{\Delta_{2}}\right]+1,\end{split}$

so the solution $x_{n}$ starting in each interval gets out of it in less than
$N_{1}$ (respectively, $N_{2}$) steps, for each $n\in\mathbb{N}$. Now we
follow the steps of the proof of Lemmata 2.8 and 6.1. We put

$\alpha_{*}:=\alpha+\ell(1-\varepsilon),\quad\alpha^{*}:=\alpha+\ell,\quad\mbox{which
implies}\quad\underline{\beta}_{2}<\beta_{3}<\alpha_{*}<\alpha^{*}<\bar{\beta}_{2}.$

Define $k_{0}=k_{0}(\alpha_{*})$ by (2.7) and note that, since
$\alpha_{*}>\beta_{3}$ and by definition (6.14) of $\beta_{3}$ (see also
(2.6)), we have $k_{0}(\alpha_{*})<\infty$. Assume that $k_{0}\geq 2$. Let
$N:=k_{0}(\alpha_{*})(N_{1}+N_{2})$. By Assumption 2, we have
$\mathbb{P}\\{\alpha+\ell\xi_{n}>\alpha+\ell(1-\varepsilon)=\alpha_{*}\\}>0$.
Applying Lemma 2.11, we conclude that there exists a random moment
$\mathcal{N}$, s.t., with probability 1, for $N$ steps in a row, starting from
$\mathcal{N}$,

$\alpha_{n}=\alpha+\ell\xi_{n}>\alpha_{*}>\beta_{3},\quad n\in
U_{\mathcal{N}}:=\\{\mathcal{N},\mathcal{N}+1,\dots,\mathcal{N}+N\\}.$

This moment $\mathcal{N}$ can be chosen greater than any other random moment
$\mathcal{M}$. To specify $\mathcal{M}$ in this part of the proof, we assume
$x_{0}\in(K_{2\tau},\,K_{2\tau+1})$ and define

$\mathcal{M}_{1}:=\inf\left\\{i\in\mathbb{N}:x_{i}\in\left(K_{2\theta+1},\,\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}\right)\right\\},$
$\mathcal{M}_{2}:=\inf\\{i>\mathcal{M}_{1}:x_{i}\in(K_{2\tau},\,K_{2\tau+1})\\},$
(6.19)

and, inductively, for $s\in\mathbb{N}$,

$\begin{split}&\mathcal{M}_{2s+1}:=\inf\\{i>\mathcal{M}_{2s}:x_{i}\in(K_{2\theta+1},\,\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\})\\},\\\
&\mathcal{M}_{2s+2}:=\inf\\{i>\mathcal{M}_{2s+1}:x_{i}\in(K_{2\tau},\,K_{2\tau+1})\\}.\end{split}$

If $\mathbb{P}\left(\Omega_{1}\right)>0$, where
$\Omega_{1}=\\{\omega\in\Omega:\mathcal{M}_{i}<\infty\,\,\mbox{for
all}\,\,i\in\mathbb{N}\\}$, it means that, with non-zero probability, a
solution circulates infinitely many times between $(K_{2\tau},\,K_{2\tau+1})$
and $(K_{2\theta+1},\,\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\})$. Recall
that by Remark 8 circulation is possible only between those intervals. We are
going to show that it is impossible for the control $\alpha$ and the noise
level $\ell$ chosen as in (6.15).

Let $\mathcal{M}:=\mathcal{M}_{2}$ be defined as in (6.19), and
$\mathcal{N}>\mathcal{M}_{2}$ as described above. Assuming that
$\mathbb{P}\left(\Omega_{1}\right)>0$, we get
$x_{n}\in(K_{2\tau}+\sigma,\,K_{2\tau+1})\cup(K_{2\theta+1},\,\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\}-\sigma)$
for all $n\geq\mathcal{N}\geq\mathcal{M}_{2}$, on $\Omega_{1}$. The solution
can get larger than $K_{2\tau+1}-\delta$ in no more than $N_{1}$ steps. If it
gets into $(K_{2\tau+1}-\delta,K_{2m+3})$, it stays there by definition of
$\delta$ (since $K_{2\tau+1}-\delta<d_{1}(\alpha-\ell)$), (2.6) and since
$\alpha_{n}\geq\alpha-\ell>\underline{\beta}_{2}$ for all $n\in\mathbb{N}$.
This cannot happen with non-zero probability on $\Omega_{1}$, see also Remark
8. Since $\alpha_{n}\geq\alpha_{*}>\beta_{3}$, for any $n\in U_{\mathcal{N}}$,
we have
$\max_{x\in[K_{2\tau},K_{2\tau+1}]}G(\alpha_{*},x)<\hat{c}(\alpha_{*})$
(another case from (6.14) is treated similarly), which implies
$\max_{x\in[K_{2\tau},K_{2\tau+1}]}G(\alpha_{*},x)<c_{k_{1}}(\alpha_{*})$ for
some $k_{1}\leq k_{0}$. Since $G(\alpha_{n},x)<G(\alpha_{*},x)$ on
$(K_{2\theta+1},\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\})$, if $x_{n}$
gets over $K_{2\theta+1}$ it satisfies $x_{n}\leq c_{k_{1}}(\alpha_{*})$.
Also, $x_{n}>K_{2\theta+1}+\delta$, since if
$x_{n}\in(K_{2\theta+1},K_{2\theta+1}+\delta)$ it stays in
$(K_{2m+1},K_{2\theta+1}+\delta)$, which, by definition of $\Omega_{1}$, can
happen only with the zero probability. So the solution can get below
$K_{2\tau+1}$ in $s$ steps, $s\leq N_{2}$ and then it satisfies
$x_{n+s}=G(\alpha_{n+s},x_{n+s-1})\geq G(\alpha_{*},x_{n+s-1})\geq
d_{k_{1}}(\alpha_{*})$, where $x_{n+s-1}\in(K_{2m+3},c_{k_{1}}(\alpha_{*}))$.
So the solution will reach $(K_{2\tau+1},K_{2\theta+1})$ in no more than $N$
steps. Since
$\bar{\beta}_{2}>\alpha_{n}\geq\alpha-\ell\geq\underline{\beta}_{2}$ for all
$n\in\mathbb{N}$, by Remark 8, the interval $(K_{2\tau+1},\,K_{2\theta+1})$ is
a new trap, so the solution stays there and converges to some equilibrium from
$\mathbb{K}$. The case when
$x_{\mathcal{N}}\in(K_{2\theta+1},\,\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}-\sigma\\})$
is similar.

Assume now that $x_{0}\in(K_{0},K_{2\tau})$.

If a solution gets into
$(K_{2\tau},\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2}\\})$ on some
$\Omega_{2}$ with $\mathbb{P}\left(\Omega_{2}\right)>0$, we denote
$\mathcal{M}_{01}=\\{i:x_{i}\in(K_{2m},\min\\{\tilde{G}(\beta_{0}),K_{2\theta+2})\\})\\}$,
consider $x_{n}$ with $n\geq\mathcal{M}_{01}$ and apply the above argument.
The case when a solution remains in $(K_{0},K_{2m})$ is covered by Lemma 6.1.
The case $x_{0}\in(\min\\{\tilde{G}(L),K_{2\theta+2}\\},K_{2(m+r)})$ is
similar.

All the above implies that the infinite circulation can only happen with the
zero probability, which concludes the proof. ∎

Received May 2020; revised July 2021; early access November 2021.