# Global Controllability for Quasilinear Non-negative Definite System of ODEs
and SDEs

Jasmina Djordjevic111Faculty of Mathematics and Natural Sciences, University
of Nis, Visegradska 33, 18000 Nis, Serbia, Electronic mail:
<EMAIL_ADDRESS>
Sanja Konjik222Faculty of Sciences, University of Novi Sad, Trg D.Obradovića
4, 21000 Novi Sad, Serbia, Electronic mail<EMAIL_ADDRESS>
Darko Mitrović 333Faculty of Mathematics, University of Vienna, Oskar
Morgenstern Platz-1, 1090 Wien, Austria, Electronic mail:
<EMAIL_ADDRESS>
Andrej Novak 444 Department of Physics, Faculty of Science, Bijenička cesta
32, University of Zagreb, Croatia, Electronic mail<EMAIL_ADDRESS>
A.Novak is the corresponding author.

###### Abstract

We consider exact and averaged control problem for a system of quasi-linear
ODEs and SDEs with a non-negative definite symmetric matrix of the system. The
strategy of the proof is the standard linearization of the system by fixing
the function appearing in the nonlinear part of the system, and then applying
the Leray-Schauder fixed point theorem. We shall also need the continuous
induction arguments to prolong the control to the final state which is a novel
approach in the field. This enables us to obtain controllability for
arbitrarily large initial data (so called global controllability).

MSC (2010): Primary: 34H05; Secondary: 49J15, 93C15, 60H10
Keywords: exact controllability, averaged controllability, quasi-linear ODEs,
quasi-linear SDEs, degenerate parabolic equation.

##  1 Introduction

A fundamental issue in control theory is how to choose a source term (usually
called the control) in the equations of the particular model which would
govern the system from the given initial state to the prescribed final state.
Such results clearly have a wide range of possible applications which makes
control theory one of the most attractive fields in mathematics. It is not
easy to say where the origins of the substantial mathematical treatment of the
theory lie, but one can find lots of information in standard books such as
[13, 26, 30]. Concerning the direction in the control theory that we are going
to pursue here, we mention [12, 15, 16, 31] and the references therein.

The theory is well developed for both linear and non-linear systems. The
theory started with linear systems and most of the related questions have been
answered (see e.g. [26]). Nonlinear systems have been intensively considered
recently and one can find various problems of this type in [2, 4, 14, 20, 27,
28, 29] and references therein. Let us briefly recall the related results from
the field.

The most general problem in this direction is the nonlinear system of the form

$\dot{y}(t)=f(t,y(t),u(t)),\ \ y(0)=y^{0}\in\mathbb{R}^{d},$ (1)

and the aim is to find the vector valued function $u$ which steers the system
to a given final state $y(T)=y^{T}$. This system was considered in [20] and it
has been proved that the system is locally approximate null controllable
provided that the linear operator of the system is approximately invertible
and the linear approximation to (1) is locally null controllable. A similar
problem – a quasilinear delay integrodifferential equation has been considered
e.g. in [2] (similar problems can be found also in [21, 22] and many others
chronologically subsequent papers).

The proofs of controllability of nonlinear systems essentially rely on the
linear theory and Leray-Schauder-Tikhonov fixed point type theorems. Roughly
speaking, we consider the Taylor expansion of $f$ in the neighborhood of
$(t,y,u)=(t,0,0)$:

$f(t,y,u)=f(t,0,0)+f_{y}(t,0,0)y+f_{u}(t,0,0)u+g(t,y,u),$

and then we linearize (1) by considering

$\begin{split}&\dot{y}(t)=f(t,0,0)+f_{y}(t,0,0)y+f_{u}(t,0,0)u+g(t,z,u),\\\
&y(0)=y^{0}\in\mathbb{R}^{d},\ \ y(T)=y^{T}\in\mathbb{R}^{d},\end{split}$ (2)

for a fixed function $z:[0,T]\to\mathbb{R}^{d}$ of appropriate regularity. One
proves or assumes existence of the control $u$ such that (2) holds and thus
one obtains the mapping

${\cal T}(z)=y$

which adjoins the state $y$ to the previously fixed function $z$. Clearly, if
we prove existence of a fixed point for the mapping ${\cal T}$ we will prove
existence of control to (1).

In the current contribution, we shall consider an ODE system with quasilinear
non-negative definite symmetric right-hand side (sometimes called degenerate
parabolic system; see (5)). Such and similar type of equations have been
intensively considered recently in both deterministic and stochastic setting
(see [1] and references therein). In particular, in [29] the control problem
of the following form has been investigated

$\dot{y}(t)=-A(t,y(t))y(t)+B(t,y(t))u(t),\ \ y(0)=y^{0},\ \ y(T)=y^{T},$ (3)

where $(t,y)\mapsto A(t,y)\in\mathbb{R}^{d\times d}$ and $(t,y)\mapsto
B(t,y)\in\mathbb{R}^{d\times n}$ are matrix-valued functions of appropriate
regularity. In that case, the situation is more straight forward because we
can avoid the Taylor expansion and simply replace $y$ appearing in $A$ and $B$
by $z$ and then apply linear theory to get the state ${\cal T}(z):=y$ and the
control $u$ satisfying

$\dot{y}(t)=-A(t,z(t))y(t)+B(t,z(t))u(t),\ \ y(0)=y^{0},\ \ y(T)=y^{T}.$ (4)

As before, we look for the fixed point of the mapping ${\cal T}$ which
provides solution to (3).

The first obstacle that we encounter is the controllability problem formulated
in (4). In general, it is not simple to prove that for any $z$ one can find
$y$ and $u$ such that (4) holds, and it is often simply taken as an assumption
(see e.g. [2, (iii)]). However, in special situations such as the one
considered in [14], one can prove the controllability for any $z$ of
appropriate regularity. The extenuating circumstance in that case is the
assumption that the equation is of strictly parabolic type i.e. the
corresponding operator is coercive, and this enables one to obtain the result
of such a generality. The coercivity of the operator then provides an
observability result which is, as well known [26, Ch III], equivalent to the
controllability.

We also note that the result from [14] holds under the assumption that the
initial data are small enough (it is called local controllability) and that
here, we introduce a method for overcoming the latter confinement. More
precisely, by introducing a change of variables $\tau=Kt$ for $K$ large
enough, we are able to obtain estimates that provide compactness of the
mapping ${\cal T}$ in a small interval $[0,T/K]$ around zero and thus
existence of the fixed point in that interval. By continuing the procedure
beginning from $T/K$ instead of $0$, and then marching forward step by step,
we reach the final moment $T$.

The paper is organized as follows. In the next section, we introduce necessary
notions and notations and formulate precisely the results that we are going to
prove. We also provide additional historical facts. In Section 3, we prove
that problem (3) admits a solution (see Theorem 3.1), while in Section 4, we
prove the stochastic variant of the result (see Theorem 4.2). In Section 5, we
provide an application in the frame of the population dynamics. The control
function represents a source which means that by choosing appropriate $u$ in
(6) we choose the rate at which we add new individuals into the system or we
simply change living conditions so that the population increases or decreases
to the desired level. We use an iterative method combined with adaptation of
final data in order to cancel computational errors and obtain satisfactory
simulations.

##  2 Notions, notations, and formulation of the results

Let us now formally introduce the system that we are going to deal with. To
this end, we need the complete probability space $(\Omega,{\cal F},{\cal
F}_{t},{\bf P})$ with the sample space $\Omega$, the $\sigma$-algebra ${\cal
F}$, the natural filtration $\\{{\cal F}_{t}\\}_{t\in[0,T]}$ generated by the
standard $n$-dimensional ($n\geq 1$) Wiener process $W_{t}(\cdot)$, and the
probability measure ${\bf P}$. For an Euclidean space $H$ (such as
$\mathbb{R}^{d},\mathbb{R}^{n},\mathbb{R}^{d\times n}$, $\mathbb{R}^{d\times
d}$ depending on the situation), we denote the spaces:

$\displaystyle L_{{\bf P}}^{2}(\Omega;C([0,T]);H)$
$\displaystyle=\\{y:[0,T]\times\Omega\to H|\ \ \forall\omega\in\Omega,$
$\displaystyle\qquad\qquad y(\omega,\cdot)\in C([0,T];H),\,{\rm
and}\,\int_{\Omega}\sup\limits_{t\in[0,T]}|y(t,\omega)|^{2}d{\bf
P}(\omega)<\infty\\},$ $\displaystyle L_{{\bf P}}^{2}(\Omega;L^{2}([0,T]);H)$
$\displaystyle=\\{y:[0,T]\times\Omega\to
H|\,\int_{\Omega}\int_{[0,T]}|y(t,\omega)|^{2}dtd{\bf P}(\omega)<\infty\\}.$

To simplify the notation, in the sequel we imply

$L^{2}(\Omega;C([0,T])):=L_{{\bf P}}^{2}(\Omega;C([0,T]);H),\qquad
L^{2}(\Omega;L^{2}([0,T])):=L_{{\bf P}}^{2}(\Omega;L^{2}([0,T]);H).$

The most general form of the system to be considered here is

$\displaystyle
dy(t,\omega)=(-A(\omega,t,y(t,\omega))y(t,\omega)+B(\omega,t,y(t,\omega))u(t,\omega))dt+Z(t,\omega)dW(t),$
(5)

where

  * •

$y(t,\omega):[0,T]\times\Omega\to\mathbb{R}^{d}$, $y\in
L^{2}(\Omega;C([0,T]))$, is adapted with respect to the filtration ${\cal
F}_{t}$;

  * •

$Z(t,\omega):[0,T]\times\Omega\to\mathbb{R}^{d\times n}$, $Z\in
L^{2}(\Omega;L^{2}([0,T]))$, is adapted with respect to the filtration ${\cal
F}_{t}$;

  * •

$u(t,\omega):[0,T]\times\Omega\to\mathbb{R}^{n}$, $u\in
L^{2}(\Omega;L^{2}([0,T]))$, is adapted with respect to the filtration ${\cal
F}_{t}$.

The function $y$ is usually called the state of the system, while $u$ is the
control. The pair $(y,Z)$ is uniquely determined via the corresponding
backward differential equation (see [5]) which means that we are actually
looking only for $y$ and $u$. Let us remark that this is both natural and
intuitive since $Z$ is random and cannot be controlled by initial and/or final
data.

Now we are ready to formally introduce definitions of controllability that we
shall need here.

###### Definition 2.1

We say that the system (5) is exactly controllable if we can find the function
$u\in C([0,T];\mathbb{R}^{n})$ such that solution $y\in
C^{1}([0,T];\mathbb{R}^{d})$ to (5) satisfies

$y(0)=y^{0},\qquad y(T)=y^{T}$

for any $y^{0},y^{T}\in\mathbb{R}^{d}$.

We note that we shall prove the exact controllability in the deterministic
case i.e. in the case when (5) does not depend on $\omega\in\Omega$.

###### Definition 2.2

We say that the system (5) admits the averaged control if we can find $u\in
L^{2}(\Omega;L^{2}[0,T])$, such that $(y,Z)$, $y\in L^{2}(\Omega;C([0,T]))$,
$Z\in L^{2}(\Omega;L^{2}[0,T])$, solves (5), and the following initial and
final conditions are satisfied:

$y(0)=y^{0},\qquad E(y(T,\cdot))=y^{T}$ (6)

where $y^{0}$ is an $d$-dimensional random variable, $y^{T}$ is a constant
vector ($E$ is the expectation operator). In this case system (5), (6), is
called controllable in average.

Although we consider a system of ODEs, the techniques used here can be
transferred to parabolic PDEs via the Galerkin approximation (this will be a
subject of future research). Lack of strict parabolicity is a major obstacle
in the proof of the solvability of (2) so we shall need to assume it below.
However, in some important special situations (see [14] and Proposition 2.3),
we are able to overcome this obstacle. More importantly, unlike the situation
from [14], we are able to prove the controllability result for (5) for
arbitrary large initial data of appropriate regularity.

Let us now proceed with the assumptions that we shall need. We shall imply:

  * (i)

$A:\Omega\times\mathbb{R}^{+}\times\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ is
a smooth non-negative definite symmetric matrix-valued process;

  * (ii)

$B:\Omega\times\mathbb{R}^{+}\times\mathbb{R}^{d}\ \to\mathbb{R}^{d\times n}$
is a matrix-valued $L^{\infty}$-uniformly bounded process;

  * (iii)

For every $v=(v_{1},\dots,v_{d})$, $v_{j}\in L^{2}(\Omega;C([0,T]))$,
$j=1,\dots,d$, the Gramian (note the difference in the notation of the final
time $T$ and the transpose ${\rm T}$ below)

$G_{c}(0,T)=E\left(\int_{0}^{T}e^{\int_{0}^{t}-A(\omega,t,v)dt^{\prime}}B(\omega,t,v)B(\omega,t,v)^{\rm
T}\left(e^{\int_{0}^{t}(-A(\omega,t,v))dt^{\prime}}\right)^{\rm T}dt\right)$
(7)

is invertible.

In the deterministic case we have $Z\equiv 0$ and we do not have dependence on
the variable $\omega$. Condition (iii) thus becomes:

  * (iii)’

If in (5) we have $Z\equiv 0$ and the coefficients are independent of
$\omega\in\Omega$, then we assume

$G_{c}(0,T)=\int_{0}^{T}e^{\int_{0}^{t}-A(t,v)dt^{\prime}}B(t,v)B(t,v)^{\rm
T}\left(e^{\int_{0}^{t}(-A(t,v))dt^{\prime}}\right)^{\rm T}dt$ (8)

is invertible for every $v\in C([0,T])$.

We note that the invertibility of the Gramian corresponding to a linear
controllability system is necessary and sufficient for the controllability to
hold (see [33, Theorem 3.2] in the stochastic and [26] in the deterministic
case). Here, urged by the nonlinearity of the system and motivated by the
construction of the proof, we needed to modify the condition in the form given
in (iii). Although condition (iii) seems too strong in the sense that it is
highly non-trivial to verify, it is a standard fare in this kind of research
(see again [2, (iii)]). However, in some interesting and important special
cases, one can simplify the condition. In [14], the controllability has been
proved for strictly parabolic equations (even partial differential equations).
Here, we shall prove controllability for a porous media type equation (which
is an example of a degenerate parabolic equation; see [9, 24]):

$y^{\prime}(t)=-A|y(t)|^{m}y(t)+Bu(t),\qquad y(0)=y^{0},$ (9)

where $A\in\mathbb{R}^{d\times d}$ and $B\in\mathbb{R}^{d\times n}$ are
constant matrices. We shall pay special attention to the equation in Section 5
where we shall introduce a numerical procedure and provide corresponding
computer simulations.

Interestingly enough, controllability for (9) is actually equivalent to the
Kalman rank condition. Let us prove this fact before we continue with the
introduction.

###### Proposition 2.3

Let $A\in\mathbb{R}^{d\times d}$ and $B\in\mathbb{R}^{d\times m}$ be constant
matrices. If the Kalman rank condition

${\rm rank}\big{[}B\big{|}AB\big{|}\dots\big{|}A^{d-1}B\big{]}=d$ (10)

holds, then the Gramian corresponding to (9)

$G_{c}(0,T)=\int_{0}^{T}e^{\int_{0}^{t}(-|v|^{m}A)dt^{\prime}}BB^{\rm
T}\left(e^{\int_{0}^{t}(-|v|^{m}A)dt^{\prime}}\right)^{\rm T}dt$ (11)

is invertible for every $v\in C([0,T])$ (in other words, (iii’) holds).

###### Remark 2.4

Let us note that here we do not require non-negative definitness or
symmetricity of the matrix $A$. However, it is necessary for the proof of
global controllability of (9).

Proof. Invertibility of (7) is equivalent to the complete controllability of
the system

$\displaystyle y^{\prime}(t)=-A|v(t)|^{m}y(t)+Bu(t)$ (12) $\displaystyle
y(0)=y^{0},\ \ y(T)=y^{T}$ (13)

for any $y^{0},y^{T}\in\mathbb{R}^{d}$ (see [26, p. 77]). In other words, if
(8) is invertible, then for every $y^{0},y^{T}$ we can find $y$ and $u$ so
that (12), (13) are satisfied and vice versa. Thus, to prove the proposition,
it is enough to show that there exist $y$ and $u$ such that (12), (13) hold.
To this end, we consider the following perturbation of (12)

$y_{\varepsilon}^{\prime}(t)=(-A|v(t)|^{m}-\varepsilon
A)y_{\varepsilon}(t)+Bu_{\varepsilon}(t),\ \ \varepsilon>0.$ (14)

We shall show that if the Kalman rank condition is satisfied then (14) is
completely controllable i.e. for every $y^{0},y^{T}$ we can find
$y_{\varepsilon}$ and $u_{\varepsilon}$ so that (14), (13) are satisfied.
Consider the Gramian corresponding to (14):

$G_{\varepsilon}(0,T)=\int_{0}^{T}e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}BB^{\rm
T}\left(e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}\right)^{\rm T}dt.$

Now, let us prove that invertibility of $G_{\varepsilon}$ is equivalent to
(10).

By contradiction, assume that $G_{\varepsilon}(0,T)$ is not invertible. This
means that there exists $w\in\mathbb{R}^{d}\backslash\\{0\\}$ such that

$\begin{split}&w\cdot G_{\varepsilon}(0,T)\cdot w^{\rm
T}=0\implies\int_{0}^{T}w\cdot
e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}BB^{\rm
T}\left(e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}\right)^{\rm T}\cdot
w^{\rm T}dt=0\\\ &\implies\int_{0}^{T}|w\cdot
e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}B|^{2}dt=0\implies w\cdot
e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}B\equiv 0\end{split}$ (15)

for every $t\in[0,T]$. From (15) we get

$w\cdot e^{-\int_{0}^{t}(|v|^{m}+\varepsilon)Adt^{\prime}}B=0\implies
w\cdot\left(I+\sum\limits_{j\in\mathbb{N}}A^{j}\frac{(-\int_{0}^{t}(|v|^{m}+\varepsilon)dt^{\prime})^{j}}{j!}\right)B=0.$

Since $\int_{0}^{t}(|v|^{m}+\varepsilon)dt^{\prime}>0$ and since $t\geq 0$ is
arbitrary, we conclude

$w\cdot A^{j}B=0,\ \ \forall j\in\mathbb{N},$

implying that the Kalman condition is not satisfied. Let us remark in passing
that according to the Cayley–Hamilton theorem, $k>n$-th power of the matrix
$A$ can be represented as a linear combination of powers of order
$0,1,\dots,n$ and thus the opposite statement holds as well.

From the above, we see that the Kalman condition implies invertibility of the
Gramian $G_{\varepsilon}(0,T)$ independently of $v$. Thus, we see that (14) is
completely controllable. Moreover, we know that the state function
$y_{\varepsilon}$ and the control $u_{\varepsilon}$ are uniformly bounded with
respect to $\varepsilon$ (see [26, p.77]). Thus, the families
$(y_{\varepsilon})$ and $(u_{\varepsilon})$ admit weak-$\star$ limits along a
common subsequence which we denote by $y$ and $u$, respectively. Clearly, the
(vector) functions $y$ and $u$ satisfy (12), (13) from where invertibility of
Gramian (11) follows. $\Box$

In the stochastic variant of (9) (to be considered numerically in Section 5),
we aim for the average control [31] and in this case the Kalman rank condition
should be modified as in [31, Theorem 1]. The proof, however, goes along the
same line as the one of Proposition 2.3. Moreover, we propose a numerical
procedure which seems quite robust and converges very fast (see Section 5) and
whose stability can be used as a practical test of the controllability.

Continuing in this direction, the second aim of the paper is to examine
controllability of the system (5) in the sense that we are looking for the
control (vector) function $(u,Z)$ which steers the state function $y$ from the
given initial state to the given final state in the sense of expectation (see
(6) below). A similar problem has been considered in [8, 33, 34], but with
linear coefficients. We also note that it is possible to include more general
SDEs as in [33], but the problem is essentially reduced to the linear variant
of the system of the form (5) (see [33, Theorem 3.4]).

We conclude this section by noting that the general stochastic case of the
described control problem (5) refers to the theory of backward stochastic
differential equations (BSDEs). BSDEs is a stochastic differential equation of
the following form

$dy(t)=f(t,y(t),z(t))dt+z(t)dW(t),\quad y(T)=\xi,$

where the function $f$ is the generator and $\xi$ is the final condition. The
solution of this equation consists of an ordered pair of adapted processes
$(y,z)$ that satisfies this equation. In the field of stochastic control, the
adapted process $z(t)$ is considered as a process of control, and the process
$y(t)$ is the process of the state of the system. The goal is to define an
adapted process $z(t)$ so that the state of the system $y(t)$ is brought to a
fixed value of $\xi$ at the time $t=T$. This problem is called the
reachability problem.

Linear type of BSDEs was introduced in [3], and later [18, 23] some results on
the existence and uniqueness of the adapted solutions in the nonlinear case
are established. Results regarding problems of perturbations for different
type of BSDEs, under several conditions are given in [5, 6, 7]. Interestingly,
the BSDE given above in general uniquely determines the functions $y$ and $z$.
Therefore, if we add the initial condition as in (5), we need another function
(the function $u$) to control the process.

##  3 A control problem for nonlinear system of ODEs

We are concerned with a control problem for the nonlinear system of ODEs of
the following form

$\displaystyle\frac{dy}{dt}(t)$ $\displaystyle=$
$\displaystyle-A(t,y(t))y(t)+B(t,y(t))u(t),\quad t\in[0,T],$ (16)
$\displaystyle y(0)$ $\displaystyle=$ $\displaystyle y^{0},$ (17)

where $A:[0,T]\times\mathbb{R}\to\mathbb{R}^{d\times d}$ is a smooth positive
semi-definite matrix-valued mapping,
$B:[0,T]\times\mathbb{R}\to\mathbb{R}^{d\times n}$ is a smooth matrix-valued
mapping uniformly bounded with respect to $L^{\infty}$-norm, and
$y^{0}\in\mathbb{R}^{d}$ is given initial state. The problem consists of
determining a function $u\in L^{2}([0,T];\mathbb{R}^{n})$ such that for the
given final state $y^{T}\in\mathbb{R}^{d}$ and the final time $T$

$y(T)=y^{T}.$ (18)

###### Theorem 3.1

Assume that conditions (i), (ii), and (iii) are satisfied. Then, there exist
unique functions $y\in C^{1}([0,T];\mathbb{R}^{d})$ and $u\in
C([0,T];\mathbb{R}^{n})$ such that the exact control problem (16), (17), (18)
is satisfied.

Proof. Our approach in solving this problem will consist of several steps.
First, we shall linearize equation (16) as follows

$\frac{dy}{dt}(t)=-A(t,v(t))y(t)+B(t,v(t))u(t),\quad t\in[0,T],$ (19)

where $v\in C([0,T])$ is fixed. For such a fixed $v$, we look for the control
$u$ steering the system from the initial state $y^{0}$ to the prescribed final
state $y^{T}$.

Since the problem is linear, we can associate to (19) an adjoint system in
order to use the observability concept and results that are well known for the
linear case [26] and even in the stochastic setting [33, Theorem 3.2]. Once
the linearized problem is solved, we shall employ fixed point arguments in
order to reach a solution to the initial problem.

The adjoint system associated to the linearized control problem is given by

$\displaystyle-\frac{d\varphi}{dt}(t)$ $\displaystyle=$
$\displaystyle-A^{*}(t,v(t))\varphi(t),\quad t\in[0,T],$ (20)
$\displaystyle\varphi(T)$ $\displaystyle=$ $\displaystyle\varphi^{T},$ (21)

where $\varphi^{T}\in\mathbb{R}^{d}$ minimizes the convex functional (see
[19])

$J(\varphi^{T})=\int_{0}^{T}|B^{*}(t,v(t))\varphi(t)|^{2}dt+\langle
y^{0},\varphi(0)\rangle-\langle y^{T},\varphi^{T}\rangle.$ (22)

Then, the so called observability condition reads

$\|\varphi^{0}\|^{2}\leq C\int_{0}^{T}\|B^{*}(t,v(t))\varphi(t)\|^{2}\,dt,$
(23)

for some constant $C$. It provides solvability of problem (19), (17), (18).
The standard proof (e.g. [33] Theorem 4.1.) provides that the controllability
of system (19) in $[0,T]$ is equivalent to observability (23) of system
(20),(21) on $[0,T]$. Moreover, a solution to problem (19) is then given by
$u(t)=B^{*}\varphi(t)$ and we have

$\begin{split}|u(t)|&=\big{|}B^{*}(t,v(t))\varphi(t)\big{|}=\big{|}B^{*}(t,v(t))e^{\int_{t}^{T}-A(s,v(s))\,ds}\varphi^{T}\big{|}\\\
&\leq|B^{*}(t,v(t))\varphi^{T}|\leq C|\varphi^{T}|,\end{split}$ (24)

where $\varphi$ is a solution to the corresponding adjoint problem.

It is also well known [26] that condition (iii) from the introduction provides
existence of the exact control for (19), (17), (18).

Introduce now a change of variables $\tau=K_{1}t$ into (19), for some
parameter $K_{1}>0$. Denote by

$\displaystyle\tilde{v}(\tau)=v\Big{(}\frac{\tau}{K_{1}}\Big{)},\ \
\tilde{u}(\tau)=u\Big{(}\frac{\tau}{K_{1}}\Big{)},\ \
\tilde{y}(\tau)=y\Big{(}\frac{\tau}{K_{1}}\Big{)},$
$\displaystyle\qquad\qquad\tilde{A}(\tau,\tilde{v}(\tau))=A(\frac{\tau}{K_{1}},\tilde{v}(\tau)),\
\ \tilde{B}(\tau,\tilde{v}(\tau))=B(\frac{\tau}{K_{1}},\tilde{v}(\tau)).$

Then, using the fact that
$\partial_{t}y=\partial_{\tau}\tilde{y}\,\frac{\partial\tau}{\partial
t}=\partial_{\tau}\tilde{y}\cdot K_{1}$, we see that (19) becomes

$K_{1}\frac{d\tilde{y}}{d\tau}(\tau)=-\tilde{A}(\tau,\tilde{v}(\tau))\tilde{y}(\tau)+\tilde{B}(\tau,\tilde{v}(\tau))\tilde{u}(\tau),\quad\tau\in[0,K_{1}T],$

or, after dividing this equation by $K_{1}$, the problem reduces to

$\displaystyle\frac{d\tilde{y}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{A}(\tau,\tilde{v}(\tau))}{K_{1}}\tilde{y}(\tau)+\tilde{B}(\tau,\tilde{v}(\tau))\frac{\tilde{u}(\tau)}{K_{1}},\quad\tau\in[0,K_{1}T],$
(25) $\displaystyle\tilde{y}(0)$ $\displaystyle=$ $\displaystyle y^{0},\ \
\tilde{y}(K_{1}T)=y^{T}.$ (26)

Next, introduce an operator ${\cal T}:{\cal C}([0,{T}])\to{\cal C}([0,{T}])$,
which maps $\tilde{v}$ to a solution $\tilde{y}$ of (25), (26) (more precisely
to the first component of the solution $(\tilde{y},\tilde{u})$) i.e.,

${\cal T}(\tilde{v})=\tilde{y}.$

Furthermore, the solution of (25), (26) is given by

$\tilde{y}(\tau)=\exp\Big{(}-\int_{0}^{\tau}\frac{\tilde{A}(s,\tilde{v}(s))}{K_{1}}\,ds\Big{)}\left[\int_{0}^{\tau}\exp\Big{(}\int_{0}^{s}\frac{\tilde{A}(\theta,\tilde{v}(\theta))}{K_{1}}\,d\theta\Big{)}\tilde{B}(s,\tilde{v}(s))\frac{\tilde{u}(s)}{K_{1}}\,ds+y^{0}\right].$
(27)

Since an arbitrary $\tilde{v}\in{\cal C}([0,K_{1}T])$ is bounded, it follows
from (24) that $\tilde{u}$ is also bounded. Thus for any $\tilde{v}$ one can
choose a constant $K_{1}\geq 1$ such that

$\Big{|}\frac{\tilde{A}(\cdot,\tilde{v}(\cdot))}{K_{1}}\Big{|}\leq
1\quad\mbox{ and
}\quad\Big{|}\tilde{B}(\cdot,\tilde{v}(\cdot))\frac{\tilde{u}}{K_{1}}\Big{|}\leq
1.$

Next, since $A$ is symmetric, it holds

$|\exp(A)|=\exp({|A|})=\exp(\max\limits_{i}\lambda_{i})$

where $\lambda_{i}$ are non-negative eigenvalues of the matrix $A$ (the
equality holds for the $2$-norm but since all norms are equivalent in finite
dimensional spaces, we will keep writing $|\cdot|$). From here and (27), one
can estimate $\tilde{y}$ as follows

$|\tilde{y}(\tau)|\leq\big{|}\exp\Big{(}-\int_{0}^{\tau}\frac{\tilde{A}(s,\tilde{v}(s))}{K_{1}}\,ds\Big{)}\big{|}\left[\int_{0}^{\tau}\big{|}\exp\Big{(}\int_{0}^{s}\frac{\tilde{A}(\theta,\tilde{v}(\theta))}{K_{1}}\,d\theta\Big{)}{\bf
1}\big{|}\,ds+|y^{0}|\right]\leq\tau+|y^{0}|,$ (28)

where ${\bf 1}=(1,\dots,1)\in\mathbb{R}^{d}$. This estimate together with (25)
implies

$|\partial_{\tau}\tilde{y}|\leq\tau+|y^{0}|+1.$ (29)

From here, we see that for any $\tilde{T}>0$ (and in particular for
$\tilde{T}=T$), the derivative $\partial_{\tau}\tilde{y}$ is uniformly bounded
on $[0,\tilde{T}]$ independently of $K_{1}$. Therefore, by the Arzela-Ascolli
theorem, the operator ${\cal T}:C([0,T])\to C([0,T])$ is compact. Moreover,
the set

$\\{x\in\mathbb{C}([0,T]):\,x=\lambda Tx,\lambda\in[0,1]\\}$

is bounded due to (28). The latter implies that the conditions of the Leray-
Schauder fixed point theorem [10] are satisfied for the operator ${\cal T}$
which in turn implies existence of a fixed point $\tilde{y}$ of ${\cal T}$
that satisfies

$\frac{d\tilde{y}}{d\tau}=-\frac{\tilde{A}(\tau,\tilde{y}(\tau))}{K_{1}}\tilde{y}(\tau)+\tilde{B}(\tau,\tilde{y}(\tau))\frac{\tilde{u}(\tau)}{K_{1}},\quad\tau\in[0,T].$

By reintroducing the change of variables $t=\tau/K_{1}$ we obtain the solution
$y_{1}(t)=\tilde{y}(t/K_{1})$ to (16) on the interval $[0,T/K_{1}]$.

We now repeat the whole procedure for the problem (16) with the initial data
given at $T/K_{1}$, i.e., $y(T/K_{1})=y_{1}(T/K_{1})$, and the same final
state $y(T)=y^{T}$. We thus obtain the function $y_{2}$ representing the
solution to (16) on the interval $[T/K_{1},T/K_{1}+T/K_{2}]$. Continuing the
procedure, we obtain a sequence of functions $y_{n}(t)$ satisfying (16) on the
intervals
$[\sum\limits_{i=0}^{n-1}T/K_{i},\sum\limits_{i=0}^{n-1}T/K_{i}+T/K_{n}]$,
where we imply $K_{0}=\infty$. Next, note that the function

$Y^{n}(t)=y_{j}(t),\ \
t\in\big{[}\sum\limits_{i=0}^{j-1}T/K_{i},\sum\limits_{i=0}^{j}T/K_{i}\big{]},\
\ j\in\\{1,\dots,n\\}$ (30)

solves equation (16) on the interval $[0,\sum\limits_{i=0}^{n}T/K_{i}]$. In
that way, we obtain a sequence of solutions $Y^{j}$ to (16) defined on
intervals $[0,\sum\limits_{i=0}^{j}T/K_{i}]$.

It remains to be shown that a solution obtained by this construction will
eventually reach the final state $y^{T}$ at $t=T$. We shall use a continuous
induction argument [11]. To this end, assume that the maximal interval on
which we can find a function $Y$ satisfying (16) and (30) is $[0,T_{y}]$ with
$T_{y}$ strictly less than $T$. Then, by repeating once again our procedure
with the inital data given at $T_{y}$, we obtain a solution defined at
$[T_{y},T_{y}+\frac{T}{K_{y}}]$ \- a contradiction with the maximality
assumption. Therefore, we obtain a desired solution $y$ to (16), (17), (18).

The solution is unique since the function $u$ is defined via the solution of
the backward problem (20), (21) (see [23]) which uniquely depends on the final
data. On the other hand, the final datum $\varphi^{T}$ is a minimizer of the
convex functional (22) and therefore it is unique. $\Box$

##  4 Stochastic nonlinear averaged control problem

In this section, we shall consider the average control problem (Definition
1.1) for a nonlinear SDE. More precisely, we aim to find $(y,u,Z)$ adapted
with respect to the filtration ${\cal F}_{t}$ such that following equation and
conditions are satisfied,

$\displaystyle{dy}$
$\displaystyle=\left(-A(\omega,t,y(t,\omega))y(t,\omega)+B(\omega,t,y(t,\omega))u(t,\omega)\right)dt+Z(t,\omega)dW(t),\quad
t\in[0,T]$ (31) $\displaystyle\qquad y(0,\omega)$
$\displaystyle=y^{0}(\omega),\ \ E(y(T,\cdot))=y^{T}.$ (32)

As in the previous section, we first need existence of control for the linear
variant of (31) (under conditions (i), (ii), and (iii)):

$\displaystyle{dy}$
$\displaystyle=\left(-A(\omega,t,v(t,\omega))y(t,\omega)+B(\omega,t,v(t,\omega))u(t,\omega)\right)dt+Z(t,\omega)dW(t),\quad
t\in[0,T]$ (33) $\displaystyle y(0,\omega)$ $\displaystyle=y^{0}(\omega),\ \
E(y(T,\cdot))=y^{T},$ (34)

for the stochastic process $v\in L^{2}(\Omega;C([0,T])$. The result is proved
in [33, Section 3.2] under the stronger requirement of exact controllability.
However, the result which can be found there does not contain an explicit
bound for the control $u$ which we need here in order to repeat the arguments
from the previous section. Therefore, we briefly recall the proof of the
control problem (33), (34).

First, let us show that without loss of generality we can assume that
$y^{0}=0$ in (32). Indeed, we solve (33) with $u\equiv 0$ to obtain the
solution $y_{1}$ to the (forward) stochastic differential equation (33)
augmented with the initial data $y_{1}(0,\omega)=y^{0}$. Then, we find the
functions $y_{2}$ (the state function) and $(u,Z)$ (the control) which solve
the control problem for (33) with the zero initial data and the final
condition $E(y_{2}(T,\cdot))=y^{T}-E(y_{1}(T,\cdot))$. The function
$y=y_{1}+y_{2}$ and the control $(u,Z)$ then represent solution to (33), (34).
Thus, as we can see, the control problem actually reduces to dealing with the
situation when $y^{0}=0$. We have the following theorem

###### Theorem 4.1

Under conditions (i), (ii) and (iii), and the assumption $y^{0}=0$, the system
(33) is controllable in average (Definition 1.1) in the sense that (34) holds
for any $y^{T}\in\mathbb{R}^{m}$. Moreover, $y\in L^{2}(\Omega;C([0,T)])$ and
$(u,Z)\in L^{2}(\Omega;L^{2}([0,T)])$ satisfying (33), (34) are unique.

Proof. Existence of the control such that (33), (34) are satisfied can be
found in [33, Section 3.6] even in the case of exact controllability. Thus, we
need to prove the second part of the theorem. To this end, recall that the
controllability in the sense (33), (34) equivalent to the following
observability inequality:

$C\int_{0}^{T}E[(B^{*}\varphi)^{2}+|\tilde{Z}|^{2}]dt\geq\delta\big{|}\varphi^{T}\big{|},$
(35)

where $C,\delta$ are constants, and $(\varphi,\tilde{Z})$ is the unique
adaptive solution to the BSDE

$d\varphi=A^{*}(\omega,t,v)\varphi dt+\tilde{Z}dW_{t},\ \
\varphi(\omega,T)=\varphi^{T},$ (36)

where $\varphi^{T}$ minimizes the functional

$J(\varphi^{T})=\frac{1}{2}\int_{0}^{T}\big{|}E\left(|B^{*}\varphi|^{2}+|\tilde{Z}|^{2}\right)\big{|}^{2}dt+\langle\varphi^{T},y^{T}\rangle.$
(37)

We remark that the functional is convex and thus the minimal value is unique.

Moreover, the control $u$ is given by

$u=B^{*}\varphi.$

Let us briefly recall the arguments leading to the latter conclusions.

We know that a bounded linear operator ${\bf K}$ between two Banach spaces
${\bf X}$ and ${\bf Y}$ is surjective if and only if the corresponding adjoint
operator ${\bf K}^{*}$ satisfies the coercivity inequality (see [33,
Proposition 4.1]):

$\begin{split}&\big{|}{\bf K}^{*}y^{*}\big{|}\geq\delta|y^{*}|_{{\bf Y}^{*}},\
\ y^{*}\in{\bf Y}^{*}.\end{split}$ (38)

In our situation, we set

$\begin{split}&{\bf K}:L^{2}(\Omega\times[0,T];\mathbb{R}^{d\times
n})\to\mathbb{R}^{d},\\\ &{\bf K}(u,Z)=y^{T}\end{split}$

and then aim to prove that ${\bf K}$ is surjective using (38). To this end,
from (33) we infer that

${\bf K}^{*}(\varphi^{T})=(B^{*}\varphi,\tilde{Z}(t)).$

Indeed, from (33) and (36), using the Itô formula, we get:

$\displaystyle d\langle y,\varphi\rangle-\langle y,d\varphi\rangle-\langle
Z,\tilde{Z}\rangle dt=\langle-Ay,\varphi\rangle dt+\langle Bu,\varphi\rangle
dt+\langle Z,\varphi\rangle dW(t).$

Taking the expectation of the latter expression and then integrating it over
$[0,T]$, we have

$\displaystyle\langle y^{T},\varphi^{T}\rangle-\langle
y^{0},\varphi^{0}\rangle-\int_{0}^{T}\langle y,d\varphi\rangle
dt-E\big{(}\int_{0}^{T}\langle(u,Z),(B^{*}\varphi,\tilde{Z})^{T}\rangle
dt\big{)}$ (39) $\displaystyle=\int_{0}^{T}E(\langle
y,-A^{*}\varphi\rangle)dt,$

from where, according to (36) and keeping in mind that $y^{0}=0$, we conclude

$\langle{\bf K}(u,Z),\varphi^{T}\rangle=\langle
y^{T},\varphi^{T}\rangle=\int_{0}^{T}E(\langle(u,Z),(B^{*}\varphi,\tilde{Z})^{T}\rangle)dt$
(40)

i.e. ${\bf K}^{*}(\varphi^{T})=(B^{*}\varphi,\tilde{Z})$. From here, we see
that coercivity inequality (38) for our operator ${\bf K}$ is equivalent to
(35).

Finally, recall that if the Banach spaces ${\bf X}$ and ${\bf Y}$ are
reflexive and the map $x^{*}\mapsto|x^{*}|^{2}_{{\bf X}^{*}}$ is Fréchet
differentiable and convex, then the linear mapping ${\bf K}:{\bf X}\to{\bf Y}$
is surjective if and only the functional

${\cal J}(y^{*};y)=\frac{1}{2}\big{|}{\bf K}^{*}y^{*}\big{|}^{2}_{{\bf
X}^{*}}+\langle y,y^{*}\rangle,\ \ y^{*}\in{\bf Y}^{*},$ (41)

admits a unique optimal minimum over $y^{*}$ (see [33, Proposition 4.1]). In
other words, conditions (38) and (41) are equivalent.

In our case, we take ${\bf X}=L^{2}(\Omega\times[0,T])$ and ${\bf
Y}=\mathbb{R}^{n}$. Then, we notice that (40) are actually Euler-Lagrange
equations for (41) with $y=y^{T}$. Thus, computing the first variation of
${\cal J}$ we see that the control

$u(t,\omega)=B^{*}\varphi(t,\omega),\ \ Z(t,\omega)=\tilde{Z},$

ensures that the final condition $y^{T}$ is satisfied and it minimizes the
functional ${\cal J}$.

This completes the proof. $\Box$

We can now adapt the procedure from the proof of Theorem 3.1 to obtain the
following result.

###### Theorem 4.2

Under the conditions (i),(ii), and (iii), the system (31) is controllable in
average in the sense that there exist unique $y\in L^{2}(\Omega;C([0,T)])$ and
$u,Z\in L^{2}(\Omega;L^{2}([0,T)])$ such that (32) holds for any $y^{0}\in
L^{1}(\Omega)$ and $y^{T}\in\mathbb{R}^{d}$.

Proof. As in the proof of Theorem 3.1, we introduce the change of variables
$\tau=K_{1}t$ into (31) for a constant $K_{1}>1$ and denote

$\begin{split}&\tilde{v}(\tau,\omega)=v\Big{(}\frac{\tau}{K_{1}},\omega\Big{)},\
\ \tilde{u}(\tau,\omega)=u\Big{(}\frac{\tau}{K_{1}},\omega\Big{)},\\\
&\tilde{y}(\tau,\omega)=y\Big{(}\frac{\tau}{K_{1}},\omega\Big{)},\ \
\tilde{Z}(\tau,\omega)=Z\Big{(}\frac{\tau}{K_{1}},\omega\Big{)},\ \
\tilde{W}_{\tau}=W_{K_{1}t},\\\
&\tilde{A}(\omega,\tau,\tilde{v}(\tau,\omega))=A(\omega,K_{1}t,y(K_{1}t,\omega)),\
\
\tilde{B}(\omega,\tau,\tilde{v}(\tau,\omega))=B(\omega,K_{1}t,y(K_{1}t,\omega))\end{split}$

Then, from (33) we have

$d\tilde{y}(\tau,\omega)=(-\frac{\tilde{A}(\omega,\tau,\tilde{v}(\tau,\omega))}{K_{1}}\tilde{y}+\tilde{B}(\omega,\tau,\tilde{v}(\tau,\omega))\frac{\tilde{u}}{{K_{1}}})d\tau+\frac{\tilde{Z}}{\sqrt{K_{1}}}d\tilde{W}_{\tau}$
(42)

and we augment it with the initial and final conditions

$\tilde{y}(0,\omega)=y^{0},\ \ E(\tilde{y}(K_{1}T,\cdot))=y^{T}.$ (43)

Next, for $\omega\in\Omega$, we introduce an operator ${\cal T}(\omega):{\cal
C}([0,T])\to{\cal C}([0,T])$ which maps $\tilde{v}(\cdot,\omega)$ to a
solution $\tilde{y}(\cdot,\omega)$ of (42), (43) which is given by

$\displaystyle\tilde{y}(\tau,\omega)=$
$\displaystyle\exp\Big{(}-\int_{0}^{\tau}\frac{\tilde{A}(\omega,s,\tilde{v}(s,\omega))}{K_{1}}\,ds\Big{)}\times$
(44)
$\displaystyle\times\Big{[}\int_{0}^{\tau}\exp\Big{(}\int_{0}^{s}\frac{\tilde{A}(\omega,\theta,\tilde{v}(\theta,\omega))}{K_{1}}\,d\theta\Big{)}\left(\tilde{B}(\omega,\tau,\tilde{v}(\omega,\tau))\frac{\tilde{u}}{{K_{1}}}\,d\tau+\frac{\tilde{Z}}{\sqrt{K_{1}}}d\tilde{W}_{\tau}\right)+y^{0}\Big{]}.$

We denote

${\cal T}_{\omega}(\tilde{v})=\tilde{y}$

and we aim to prove the existence of the fixed point for the mapping ${\cal
T}_{\omega}$ almost surely (i.e. for ${\bf P}$-almost every
$\omega\in\Omega$).

In order to estimate expression (44), we need to bound $\tilde{u}$. If we
notice that (39) are Euler-Lagrange equations corresponding to the
minimization problem (37), we see that $u=B^{*}\varphi$, where $\varphi$ is
the solution to (36). This is a BSDE and we can use the standard procedure to
get the necessary estimate (see [18]). Indeed, from the Itô formula for
$|\varphi(\cdot)|^{2}$, we have

$\frac{1}{2}d|{\varphi}|^{2}=\left(\langle
A^{*}\varphi,\varphi\rangle+\frac{|\tilde{Z}|^{2}}{2}\right)dt+\langle\varphi,\tilde{Z}\rangle
d{W}_{t}.$

By integrating the later expression on $(t,T)$ and finding the expectation of
the obtained expression, while taking into account the non-negative
definitness of the matrix $A^{*}$, we get

$E(|\varphi(t,\cdot)|^{2})+E(\int_{t}^{T}|\tilde{Z}(t^{\prime},\cdot)|^{2}dt^{\prime})\leq|y^{T}|^{2}.$

Thus, since $u=B^{*}\varphi$ and $\varphi$ is almost surely continuous we have
that the control $u$ is also almost surely continuous (and thus almost surely
bounded) on $[0,T]$. Moreover, according to the results from [17, Theorem
5.1], we know that the martingale part $Z$ of (33) is bounded (more precisely,
it is of cádlág type). Thus, we can choose $K_{1}\geq 1$ such that almost
surely

$\begin{split}&\Big{|}\frac{\tilde{A}(\omega,\tau,\tilde{v}(\tau,\omega))}{K_{1}}\Big{|}\leq
1,\ \ \frac{|B(\omega,s,\tilde{v}(\omega,s))\tilde{u}(s,\omega)|}{K_{1}}\leq
1\\\ &\Big{|}\frac{\tilde{u}}{K_{1}}\Big{|}\leq 1,\ \ {\rm and}\ \
\Big{|}\frac{\tilde{Z}^{2}}{K_{1}}\Big{|}\leq 1.\end{split}$ (45)

Having this in mind, we can essentially repeat the procedure from the previous
section. We square (44) and apply the expectation operator. After taking into
account the Itô isometry we get

$\begin{split}E(|\tilde{y}(\tau,\omega)|^{2})\leq&3E\Big{(}\exp\Big{(}-2\int_{0}^{\tau}\frac{\tilde{A}(\omega,s,\tilde{v}(s,\omega))}{K_{1}}\,ds\Big{)}\times\\\
&\times\left[\left(\int_{0}^{\tau}\exp\Big{(}\int_{0}^{s}\frac{\tilde{A}(\omega,\theta,\tilde{v}(\theta,\omega))}{K_{1}}\,d\theta\Big{)}\frac{|B(\omega,s,\tilde{v}(\omega,s))\tilde{u}(s,\omega)|}{K_{1}}\,ds\right)^{2}\right.\\\
&\qquad\left.+\int_{0}^{\tau}\exp\Big{(}\int_{0}^{s}2\frac{\tilde{A}(\omega,\tau,\tilde{v}(\tau,\omega))}{K_{1}}\,d\theta\Big{)}\frac{|\tilde{Z}(s,\omega)|^{2}}{K_{1}}\,ds+|y^{0}|^{2}\Big{)}\right]\\\
&\qquad\qquad\leq\bar{C}(\tau^{2}+\tau+E|y^{0}|^{2}),\end{split}$

for a constant $\bar{C}>0$. Next, we need to estimate the $\tau$-variation of
$E(|\tilde{y}(\tau,\omega))$.

To this end, we rewrite (42) in the integral form and again combine the
expectation operator with the Itô isometry. We have for any $0<\Delta t<1$

$\begin{split}&E\left(\Big{|}\tilde{y}(\tau+\Delta\tau)-\tilde{y}(\tau)\Big{|}^{2}\right)\\\
&\leq
2E\left[\left(\int_{\tau}^{\tau+\Delta\tau}\left(-\frac{\tilde{A}(\omega,s,\tilde{v}(s,\omega))}{K_{1}}\tilde{y}(s,\cdot)+\frac{B(\omega,s,\tilde{v}(\omega,s))\tilde{u}(s,\cdot)}{K_{1}}\right)ds\right)^{2}\right]\\\
&+2E\left[\int_{\tau}^{\tau+\Delta\tau}\frac{\tilde{Z}(s,\cdot)^{2}}{K_{1}}ds\right].\end{split}$

From here and (45), we have

$E\left(\frac{\Big{|}\tilde{y}(\tau+\Delta\tau)-\tilde{y}(\tau)\Big{|}^{2}}{\Delta\tau}\right)\leq
C.$

Hence, for any $\tilde{T}>0$, $|\tilde{y}|^{2}$ and its Hölder norm
$\|\tilde{y}\|_{C^{1/2}([0,T])}$ are uniformly bounded for ${\bf P}$-almost
every $\omega\in\Omega$ on $[0,\tilde{T}]$.

Thus, we can repeat the procedure from the proof of Theorem 3.1 (after
relation (29)) to conclude the existence of the fixed point for the mapping
${\cal T}_{\omega}$ for ${\bf P}$-almost every $\omega\in\Omega$. The obtained
function $y$ is the solution to (33), (34). $\Box$

##  5 An application on population dynamics

A motivation for considering system (9) was the porous media equation which
raises a constant interest from the viewpoint of both purely theoretical or
applied mathematics. It has the form

$\partial_{t}u=\nabla\big{(}(m-1)|u|^{m-1}\nabla u\big{)}$ (46)

and it describes the standard model of gas flow through a porous medium
(Darcy-Leibenzon-Muskat), nonlinear heat transfer (Zel’dovich-Raizer),
Boussinesq’s groundwater flow, population dynamics (Gurtin-McCamy) etc. [32].
Different variant of the equation such as stochastic [24] or fractional [25]
are also well known and well investigated.

To this end, we shall consider an ODE counter-part of (46). More precisely, we
shall consider population dynamics with species intending to avoid crowding.
It is interesting to inspect how to control the populations by adding new
individuals or in some other way by improving the living conditions (in the
frame of the given nonlinear model, of course). We assume that we have two
species whose population quantities are denoted by $y_{1}$ and $y_{2}$ and
which have tendency to avoid crowding. Mathematical model of the phenomenon is
given by

$\begin{split}dy_{1}&=(|y_{1}+y_{2}|(-2y_{1}+2y_{2})+u)dt+Z_{1}dW_{t}\\\
dy_{2}&=|y_{1}+y_{2}|(y_{1}-y_{2})dt+Z_{2}dW_{t}\end{split}$ (47)

and we aim to maintain the population by randomly introducing new individuals
$u_{1}$ and $u_{2}$ of the corresponding species into the system. Thus, we
start with the populations

$y_{1}(0,\omega)=y_{2}(0,\omega)=1,$

and we want to have the same population at a final time $T=0.5$

$E(y_{1}(0.5,\cdot))=E(y_{2}(0.5,\cdot))=2.$ (48)

By direct substitution, we know that for the problem

$dy=(A(t)y+B(t)u)dt+ZdW_{t},\ \ E(y(0,\cdot))=y_{0},\ \
E(y(0.5,\cdot))=y_{1},$

the control function is given by

$u(t,\omega)=-B^{\rm T}\Phi^{\rm T}(0,t)W_{c}(0,1)^{-1}[y_{0}-\Phi(0,1)y_{1}]$
(49)

where

$W_{c}(0,t)=\int_{0}^{t}\Phi(0,\tau)B(\tau)B^{\rm T}(\tau)\Phi^{\rm
T}(0,\tau)d\tau$ (50)

and $\Phi$ solves the system

$\Phi^{\prime}=A(t)\Phi,\ \ \Phi(0,0)=I.$ (51)

In order to solve (47), we use the following recursive procedure

$\begin{split}dy^{n}_{1}&=(|y^{n-1}_{1}+y^{n-1}_{2}|(-2y^{n}_{1}+2y^{n}_{2})+u^{n})dt+Z^{n}_{1}dW_{t}\\\
dy^{n}_{2}&=|y^{n-1}_{1}+y^{n-1}_{2}|(y^{n}_{1}-y^{n}_{2})dt+Z^{n}_{2}dW_{t}\end{split}$

where $u^{n}$ is given by the corresponding variant of (49). The results are
shown on Figures 2 and 3. Please note that instead of the control $u$, we plot
the scaled control $u/10$ in order to provide better presentation of the
results.

As we can see on Figure 1 showing the control $u$ and the approximate state
$(y_{1},y_{2})$ after $n=1,2,3,4,5,6$ iterations in the deterministic
situation, our method provides the exact control for the given nonlinear
system. We notice that the procedure stabilizes already after the third
iteration, for instance, the relative change of $u$ from fourth to fifth
iteration is less than $1\%$ in the $L^{\infty}$-norm.

(a) Initial iteration

(b) After two iterations.

(c) After three iterations.

(d) After four iterations.

(e) After five iterations.

(f) After six iterations.

Figure 1: The numerical procedure after one to six iterations.

We can also notice that the final states are not exactly at the prescribed
values $y_{1}(0.5)=y_{2}(0.5)=2$ which is due to the numerous computations of
various integrals (50) and matrix exponential (51). More precisely, we need to
compute $\mathcal{O}(1/dt)$ integrals in every iteration until the fixed point
is reached, so the appearance of the numerical error is not unexpected. For
instance, approximation of the matrix exponential by the polynomial of the
second order gives even worse results since in this case we end up in

$y_{1}(0.5)=1.7640,\ \ y_{2}(0.5)=2.1919.$

On the other hand, approximations of $e^{A}$ by the polynomials of fifth and
sixth order provide the final states

$\displaystyle y_{1}(0.5)=1.9851,\ \ y_{2}(0.5)=2.1195\ \ \text{ fifth order
approximation of the exponential.}$ $\displaystyle y_{1}(0.5)=2.0209,\ \
y_{2}(0.5)=2.1070\ \ \text{ sixth order approximation of the exponential.}$

Interestingly, the approximation of seventh order only slightly changes the
final states.

In order to improve the procedure, we introduce the perturbation of the final
states after each iteration as follows: after the first iteration, the reached
values of the final states are $y_{1}(0.5)=1.93$ and $y_{2}(0.5)=1.99$ while
the prescribed final state was $y_{1}(0.5)=y_{2}(0.5)=2$. For the next
iteration, we shall therefore require $y_{1}(0.5)=2+0.07\alpha$ and
$y_{2}(0.5)=2+0.01\alpha$, where $\alpha$ is the relaxation parameter and in
our experiments $\alpha=10^{-1}$. In this way, we expect to compensate for the
numerical errors by purposely introducing a perturbation of the final states.
Given that we have omitted the stochastic forcing, we are able to fine tune
the perturbation and obtain almost perfect results as demonstrated on Figure
2.

(a) Initial iteration

(b) After two iterations.

(c) After three iterations.

(d) After four iterations.

(e) After five iterations.

(f) After six iterations.

Figure 2: The numerical procedure after one to six iterations involving
perturbations of final states.

An extension of this approach to the stochastic case is shown in Figure 3,
where we have a simulation after five iterations for six different samplings.
We see that the final state is not reached exactly as the stochastic term
distorts evolution of the state functions from the initial to the final
conditions. Instead, the final state is reached in an averaged sense (48). In
order to demonstrate this, we have performed five sets of numerical experiment
with twenty different samplings of the stochastic forcing in each set, while
limiting the maximal number of iterations to $n=1$ in the first set of
experiments, $n=2$ in the second set, and continuing in a similar fashion to
$n=5$ for the last set of twenty experiments. The expected values of the final
states for each set of twenty experiments as well as the total expected value
(for all 100 experiments) are provided bellow in the Table 1. The total
expected value is the average of final states obtained for all iterations.
Although the total expected value is related to a slightly different
processes, they solve similar equations and the convergence of the method
seems to be fast.

Table 1: Expected values after $n=1,2,\ldots,5$ iterations (with twenty experiments for each value of $n$) and total expected value. Exp and Iter denote expected value and number of iterations, respectively. Exp$\setminus$ Iter | 1 iter | 2 iter | 3 iter | 4 iter | 5 iter | Total
---|---|---|---|---|---|---
$E(y_{1}(1,\cdot))$ | 1.8126 | 2.0025 | 1.8742 | 2.0805 | 2.1851 | 1.9910
$E(y_{2}(1,\cdot))$ | 1.8391 | 2.0206 | 1.9112 | 2.0942 | 2.1912 | 2.0113

Figure 3: The state evolution for six different samplings.

###### Remark 5.1

The introduced method opens new possibilities in inspecting controllability
properties of the nonlinear systems. An obvious generalization of the given
procedure can be performed on the systems of the form

$\displaystyle
dy(t)=\left(-A(t,\omega,y(t,\omega),Z(t,\omega),u(t,\omega))y(t,\omega)+B(t,\omega)u(t,\omega)+C(t,\omega)Z(t,\omega)\right]dt$
$\displaystyle\phantom{dy(t)=}+\left[D(t,\omega)y(t,\omega)+E(t,\omega)u(t,\omega)+F(t,\omega)Z(t,\omega)\right]dW(t),$
$\displaystyle\phantom{dy(t)}y(0,\omega)=y^{0},\qquad Ey(T,\omega)=y^{T},$
(52)

where $A$ is non-negative definite matrix while $B,C,D,E,F$ are well defined
matrices. Combining the approach from [33] with the method introduced here,
all statements from Section 3 should apply to this case which opens the
possibility to solve the averaged control problem for (52).

## Acknowledgement

This work is partially supported by Ministry of Education, Science and
Technological Development of the Republic of Serbia (Grant No.
451-03-9/2021-14/200125). It is also supported by the Lise Meitner project
number M 2669-N32 of the Austrian Science Fund (FWF) and by the Croatian
Science Foundation under Project MiTPDE (number IP-2018-01-2449). This article
is based upon work from COST Action CA15225 FRACTIONAL and CA15125 DENORMS
supported by COST (European Cooperation in Science and Technology).

The permanent address of D.M. is University of Montenegro, Montenegro.

## References

  * [1] A. Babiarz, J. Klamka, M. Niezabitowski, Schauder’s fixed-point theorem in approximate controllability problems, Int. J. Appl. Math. Comput. Sci. 26 (2016), 263–275.
  * [2] K. Balachanfran, R. Sakthivel, S. Marshal Anthoni, Controllability of quasilinear integrodifferential systems in Banach spaces, Nihokai Math. J. 12 (2001), 1–9.
  * [3] J. M. Bismut, Conjugate Convex Functions in Optimal Stochastic Control, J. Math. Anal. Appl. 44 (1973) 384-404
  * [4] H. Cho, F.E. Udwadia, Explicit Solution to the Full Nonlinear Problem for Satellite Formation-Keeping, Acta Astronautica, 67 (2010), 369–387.
  * [5] S. Janković, M Jovanović, J. Djordjević, Perturbed backward stochastic differential equations, Mathematical and Computer Modeling, 55 (2012), 1734–1745.
  * [6] J. Djordjević, S. Janković, Backward stochastic Volterra integral equations with additive perturbations, Applied Mathematics and Computation, 265 (2015), 903-910.
  * [7] J. Djordjević, S. Janković, Reflected backward stochastic differential equations with perturbations, Discrete and Continuous Dynamical System - A, 38(4)(2018) 1833- 1848.
  * [8] B. Gashi, Stochastic minimum-energy control, System Control Lett. 85 (2015), 70–76.
  * [9] B. Gess, Finite speed of propagation for stochastic porous media equations, SIAM J. Math. Anal. 45 (2013), 2734–2766.
  * [10] D.Gilbarg, N.Trudinger, Elliptic Partial Differential Equations of Second Order , Springer, 2001.
  * [11] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathematiques and Applications 26, Springer, 1996.
  * [12] M. Lazar, E. Zuazua, Averaged control and observation of parameter-depending wave equations, C. R. Acad. Sci.Paris, Ser. I 352 (2014), 497–502.
  * [13] J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations de Systémes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, RMA 8 (1998).
  * [14] X. Liu, X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Optim. 50 (2012), 2046–2064.
  * [15] J. Lohéac, E. Zuazua, From averaged to simultaneous controllability of parameter dependent finite-dimensional systems, Annales de la Faculté des Sciences de Toulouse Mathématiques (6), 25 (2016), 785–828.
  * [16] M. Lazar, E. Zuazua, Greedy controllability of finite dimensional linear systems, Automatica J. IFAC 74 (2016), 327–340.
  * [17] J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations, Ann. Appl. Probab. 12 (2002), 1390–1418.
  * [18] X. Mao, Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Processes and their Applications 58 (1995), 281–292.
  * [19] S. Micu and E. Zuazua, An introduction to the controllability of linear PDE, Contrôle non linéaire et applications, Sari, T., ed., Collection Travaux en Cours Hermann (2005), 67-150.
  * [20] K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM Journal on Control and Optimization, 25 (1987), 715–722.
  * [21] K. Naito, On controllability for a nonlinear Volterra equation, Nonlinear Analysis TMA 18 (1992), 99–108.
  * [22] K. Naito, J. Y Park, Approximate controllability for trajectories of a delay Volterra control system, J.Optim.Th.App. 61 (1989), 271–279.
  * [23] E. Pardoux, S.G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1) (1990) 55–61.
  * [24] G. Da Prato, M. Röckner, and V. Barbu, Stochastic Porous Media Equations, Springer, 2016
  * [25] A.de Pablo, F.Quirós, A.Rodriguez, J.L.Vazquez, A fractional porous medium equation, Advances in Mathematics 226(2011), 1378–1409.
  * [26] C. C. Remsing, Linear control, Lecture Notes, Rhodes University, Grahamstown 6140, South Africa, 2010.
  * [27] F.E.Udwadia, R. E. Kalaba, A New Perspective on Constrained Motion. Proceedings of the Royal Society of London, Series A, 439 (1992), 407–410.
  * [28] F.E.Udwadia, Optimal Tracking Control of Nonlinear Dynamical Systems, Proceedings of the Royal Society of London, Series A, 464 (2008), 2341–2363.
  * [29] X. Zhang, Remarks on the controllability of some quasilinear equations, Some problems of Nonlinear Hyperbolic Equations and Applications (2010), 437–452.
  * [30] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some results and open problems, Handbook of Differential Evolutionary Equations, vol. 3, C. M. Dafermos and E. Feireisl eds, Elsevier Science, Amsterdam, 527–621.
  * [31] E. Zuazua, Averaged control, Automatica, 50 (2014), 3077–3087.
  * [32] J. L. Vasquez, The Porous Medium Equation: Mathematical Theory, Clarendon Press, 2006.
  * [33] Y. Wang, D. Yang, J. Yong, Z. Yu, Exact controllability of Linear Stochastic Differential Equations and Related Problems, Math. Control and Related Fields 7 (2017), 305–345.
  * [34] Y.Wang, C. Zhang, The norm optimal control problem for stochastic linear control systems, ESAIM Control Optim. Calc. Var., (21) (2015), 399-413.