# Observability and State Estimation for a Class of Nonlinear Systems

John Tsinias and Constantinos Kitsos J. Tsinias is with the Department of
Mathematics, National Technical University of Athens, Zografou Campus 15780,
Athens, Greece<EMAIL_ADDRESS>(corresponding author). C. Kitsos
is with the Department of Mathematics, National Technical University of
Athens, Zografou Campus 15780, Athens, Greece and Univ. Grenoble Alpes, CNRS,
Grenoble INP∘, GIPSA-lab, 38000 Grenoble, France, email:
<EMAIL_ADDRESS>Institute of Engineering
Univ. Grenoble Alpes

###### Abstract

We derive sufficient conditions for the solvability of the state estimation
problem for a class of nonlinear control time-varying systems which includes
those, whose dynamics have triangular structure. The state estimation is
exhibited by means of a sequence of functionals approximating the unknown
state of the system on a given bounded time interval. More assumptions
guarantee solvability of the state estimation problem by means of a hybrid
observer.

###### Index Terms:

nonlinear systems, observability, state estimation, hybrid observers.

##  I Introduction

Many important approaches have been presented in the literature concerning the
state estimation problem for a given nonlinear control system (see for
instance [1] - [19], [21] - [25], [27], [28] and relative references therein).
Most of them are based on the existence of an observer system exhibiting state
estimation. The corresponding hypotheses include observability assumptions and
persistence of excitation. In [8], [9] and [10] Luenberger type observers and
switching estimators are proposed for a general class of triangular systems
under weaker assumptions than those adopted in the existing literature. In
[16] and [24] the state estimation is exhibited for a class of systems by
means of a hybrid observer.

The present note is inspired by the approach adopted in [16], where a ”hybrid
dead-beat observer” is used, as well as by the methodologies applied in [20],
[26] and [29], where, fixed point theorems are used for the establishment of
sufficient conditions for observability and asymptotic controllability. Our
main purpose is to establish that, under certain hypotheses, including
persistence of excitation, the State Estimation Design Problem (SEDP) around a
given fixed value of initial time is solvable for a class of nonlinear systems
by means of a sequence of mappings $X_{\nu}$, exclusively dependent on the
dynamics of the original system, the input $u$ and the corresponding output
$y$, and further each $X_{\nu}$ is independent of the time-derivatives of $u$
and $y$. An algorithm for explicit construction of these mappings is provided.

We consider time-varying finite dimensional nonlinear control systems of the
form:

$\displaystyle\begin{aligned} \dot{x}=A(t,y,u)x+f(t,y,x,u)\\\
(t,y,x,u)\in{{\mathbb{R}}_{\geq
0}}\times{{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{n}}\times{{\mathbb{R}}^{m}}\end{aligned}$
(1.1a) with output $\displaystyle y=C\left(t,u\right)x$ (1.1b)

where $u$ is the input of (1.1). Our main results establish sufficient
conditions for the approximate solvability of the SEDP for (1.1). The paper is
organized as follows. Section II contains definitions, assumptions, as well as
statement and proof of our main result (Proposition 2.1) concerning the state
estimation problem for the general case (1.1). We apply in Section III the
main result of Section II for the derivation of sufficient conditions for the
solvability of the same problem for certain subclasses of systems (1.1), whose
dynamics have triangular structure (Proposition 3.1). According to our
knowledge, the sufficient conditions proposed in Sections II, III are weaker
than those imposed in the existing literature for the solvability of the
observer design problem for the same class of systems. More extensions are
discussed in Section IV of present work, concerning the solvability of the
SEDP by means of a hybrid observer under certain additional assumptions
(Proposition 4.1).

Notations: For given $x\in{{\mathbb{R}}^{n}}$, $\left|x\right|\text{ }$denotes
its usual Euclidean norm. For a given constant matrix
$A\in{{\mathbb{R}}^{m\times n}}$, ${A}^{\prime}$ denotes its transpose and
$\left|A\right|:=\sup\left\\{\left|Ax\right|,\left|x\right|=1\right\\}$ is its
induced norm. For any nonempty set $I\subset\mathbb{R}$ and map $I\ni t\to
A(t)\in{{\mathbb{R}}^{m\times n}}$ we adopt the notation
$\|A(\cdot){{\|}_{I}}\text{:=}\operatorname{ess}.\sup\\{\left|A(t)\right|,t\in
I\\}$.

##  II Hypotheses and Main Result

In this section we provide sufficient conditions for observability of (1.1)
and solvability of the SEDP. We assume that for each $t\geq 0$ the mappings
$A\left(t,\cdot,\cdot\right):$
${{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{m}}\to{{\mathbb{R}}^{n\times n}}$ ,
$C\left(t,\cdot\right):{{\mathbb{R}}^{m}}\to{{\mathbb{R}}^{k\times n}}$ and
$f\left(t,\cdot,\cdot,\cdot\right):$
${{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{n}}\times{{\mathbb{R}}^{m}}\to{{\mathbb{R}}^{n}}$
are continuous and further $f$ is (locally) Lipschitz continuous with respect
to $x$, i.e., for every bounded $I\subset{{\mathbb{R}}_{\geq
0}},X\subset{{\mathbb{R}}^{n}},U\subset{{\mathbb{R}}^{m}}$ and
$Y\subset{{\mathbb{R}}^{k}}$ there exists a constant $C>0$ such that

$\displaystyle\left|f\left(t,y,{{z}_{1}},u\right)-f\left(t,y,{{z}_{2}},u\right)\right|\leq
C\left|{{z}_{1}}-{{z}_{2}}\right|,$ $\displaystyle\forall\left(t,y,u\right)\in
I\times Y\times U,\,{{z}_{1}},{{z}_{2}}\in X$ (2.1)

Also, assume that for any
$\left(x,y,u\right)\in{{\mathbb{R}}^{n}}\times{{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{m}}$
the mappings $A\left(\cdot,y,u\right)$, $f\left(\cdot,y,x,u\right)$ and
$C(\cdot,u)$ are measurable and locally essentially bounded in
$\mathbb{R}_{\geq 0}$. Let ${{t}_{0}}\geq 0,\tau>{{t}_{0}}$ and let
${{U}_{[{{t}_{0}},\tau]}}$ be a nonempty set of inputs
$u\in{{L}_{\infty}}\left(\left[{{t}_{0}},\tau\right];{{\mathbb{R}}^{m}}\right)$
of (1.1) (without any loss of generality it is assumed that
${{U}_{[{{t}_{0}},\tau]}}$ is independent of the initial state). Define by
${{Y}_{[{{t}_{0}},\tau],u}}$ the set of outputs $y$ of (1.1) defined on the
interval $[{{t}_{0}},\tau]$ corresponding to some input
$u\in{{U}_{[{{t}_{0}},\tau]}}$:

$\displaystyle Y_{[t_{0},\tau],u}:=\\{y\in
L_{\infty}\left([t_{0},\tau];\mathbb{R}^{k}\right):y(t)=C\left(t,u(t)\right)$
$\displaystyle\times x\left(t;t_{0},x_{0},u\right),\text{a.e.
}t\in[t_{0},\tau],\text{ for certain }x_{0}\in\mathbb{R}^{n}\\}$ (2.2)

provided that ${{t}_{\max}}\geq\tau$ where
$t_{\max}=t_{\max}(t_{0},x_{0},u)\leq+\infty$ is the maximum time of existence
of the solution $x(\cdot;{{t}_{0}},{{x}_{0}},u)$ of (1.1) with initial
$x({{t}_{0}};{{t}_{0}},{{x}_{0}},u)={{x}_{0}}$.

###### Definition II.1

Let $I$ be a nonempty subset of ${{\mathbb{R}}_{\geq 0}}$. We say that (1.1)
is observable over $I$, if for all ${{t}_{0}}\in I$, almost all
$\tau>{{t}_{0}}$ near ${{t}_{0}}$, input $u\in{{U}_{[{{t}_{0}},\tau]}}$ and
output $y\in{{Y}_{[{{t}_{0}},\tau],u}}$, there exists a unique
${{x}_{0}}\in{{\mathbb{R}}^{n}}$ such that

$\displaystyle y(t)=C(t,u(t))x(t;{{t}_{0}},{{x}_{0}},u),\text{a.e.
}t\in\left[{{t}_{0}},\tau\right]$ (2.3)

According to Definition 2.1, observability is equivalent to the existence of a
(probably noncausal) functional
$X(\cdot,\cdot,\cdot):\left\\{{{t}_{0}}\right\\}\times{{Y}_{[{{t}_{0}},\tau],u}}\times{{U}_{[{{t}_{0}},\tau]}}\to{{\mathbb{R}}^{n}}$,
such that, for every ${{x}_{0}}\in{{\mathbb{R}}^{n}}$ for which (2.3) holds
for certain $u\in{{U}_{[{{t}_{0}},\tau]}}$ and
$y\in{{Y}_{[{{t}_{0}},\tau],u}}$, we have:

$\displaystyle X({{t}_{0}},y,u)={{x}_{0}}$ (2.4)

and $X$ is exclusively dependent on the input $u$ and the output $y$ of (1.1)
and is in general noncausal. Knowledge of $X$ satisfying the previous
properties guarantees knowledge of the initial state value, thus knowledge of
the future values of the solution of (1.1), provided that the system is
complete.

###### Definition II.2

Let $I$ be a nonempty subset of ${{\mathbb{R}}_{\geq 0}}$. We say that the
SEDP is solvable for (1.1) over $I$, if there exists a functional
$X(\cdot,\cdot,\cdot):\left\\{{{t}_{0}}\right\\}\times{{Y}_{[{{t}_{0}},\tau],u}}\times{{U}_{[{{t}_{0}},\tau]}}\to{{\mathbb{R}}^{n}}$,
${{t}_{0}}\in I$, $\tau>{{t}_{0}}$ near ${{t}_{0}}$, being in general
noncausal, such that (2.4) is fulfilled for every
${{x}_{0}}\in{{\mathbb{R}}^{n}}$ for which (2.3) holds for certain
$u\in{{U}_{[{{t}_{0}},\tau]}}$ and further $X$ is exclusively depended on $u$
and $y$ and the dynamics of (1.1) and it does not include any differentiation
of their arguments. It turns out that $X$ is independent of the time-
derivatives of $u$ and $y$, whenever they exist.

It is worthwhile to remark here that the approach proposed in [16] for the
construction of a hybrid dead-beat observer for a subclass of systems (1.1) is
based on an explicit construction of a (noncausal) map $X$ satisfying (2.4).
However, for general nonlinear systems, the precise and direct determination
of the functional $X$ is a difficult task. The difficulty comes from our
requirements for the candidate $X$ to be exclusively dependent on $u$ and $y$
and the dynamics of system and, for practical reasons, it should be
independent of the time-derivatives of $u$ and $y$. We next provide a weaker
sequential type of definition of the solvability of SEDP, which is adopted in
the present work, in order to achieve the state determination for general case
(1.1) by employing an explicit approximate strategy.

###### Definition II.3

We say that the approximate SEDP is solvable for system (1.1) over $I$, if
there exist functionals
$X_{\nu}(\cdot,\cdot,\cdot),\nu=1,2,\ldots:\left\\{{{t}_{0}}\right\\}\times{{Y}_{[{{t}_{0}},\tau],u}}\times{{U}_{[{{t}_{0}},\tau]}}\to{{\mathbb{R}}^{n}}$(being
in general noncausal), such that, if we denote:

$\displaystyle{{\xi}_{\nu}}:={{X}_{\nu}}({{t}_{0}},y,u),\ {{t}_{0}}\in
I,y\in{{Y}_{[{{t}_{0}},\tau],u}},u\in{{U}_{[{{t}_{0}},\tau]}}$ (2.5)

then
(I) the mappings $X_{\nu}$ are exclusively dependent on the input $u$ and the
corresponding output $y$, the dynamics of system (1.1) and further their
domains do not include any differentiation of their arguments. It turns out
that each $X_{\nu}$ should be independent of the time-derivatives of $u$ and
$y$ (whenever they exist);
(II) the following hold:

$\displaystyle\lim_{\nu\to\infty}{{\xi}_{\nu}}={{x}_{0}};$ (2.6a)

It should be emphasised that uniqueness requirement in (2.6b) is not
essential. We may replace (2.6b) by the assumption that there exists $x_{0}$
satisfying both (2.3) and (2.6a). Then uniqueness of such a vector $x_{0}$ is
a consequence of (2.6a), definition (2.5) and the fact that each functional
$X_{\nu}$ exclusively depends on $u$ and $y$.

Obviously, according to the definitions above, the following implications
hold:

Solvability of SEDP $\Rightarrow$ Solvability of approximate SEDP
$\Rightarrow$ Observability (over $I$). For completeness, we note that the
first implication follows by setting $X_{\nu}:=X,\nu=1,2,\ldots$ in (2.4). The
second implication is a direct consequence of both assumptions (2.6a,b),
definition (2.5) and the exclusive dependence of each $X_{\nu}$ from $u$ and
$y$. The converse claims are not in general valid; particularly, observability
does not in general imply solvability of the (approximate) SEDP, due to the
additional requirements of Definitions 2.2 and 2.3 concerning the independence
$X$, $X_{\nu}$, respectively, from the time-derivatives of $u$ and $y$.

From (2.6a) we deduce that, if the approximate SEDP is solvable for (1.1) over
$I$, then for any $T>t_{0}$ for which
$T\leq{{t}_{\max}}({{t}_{0}},{{x}_{0}},u)$ it holds:

$\displaystyle\lim_{\nu\to\infty}\|x(\cdot;t_{0},\xi_{\nu},u)-x(\cdot;t_{0},x(t_{0}),u)\|_{[t_{0},T]}=0$
(2.7)

###### Remark II.1

(i) Condition (2.7) guarantees that for any given interval $[{{t}_{0}},T]$
with $T\leq{{t}_{\max}}({{t}_{0}},{{x}_{0}},u)$, the unknown solution
$x(s;{{t}_{0}},x({{t}_{0}}),u),s\in[{{t}_{0}},T]$ of (1.1) is uniformly
approximated by a sequence of trajectories $\hat{x}$ of the system

$\displaystyle\dot{\hat{x}}\left(t\right)=A(t,y,u)\hat{x}+f(t,y,\hat{x},u),\hat{x}({{t}_{o}})={{\xi}_{\nu}},\nu=1,2,\ldots$

with ${{\xi}_{\nu}},\nu=1,2,\ldots$ as given in Definition 2.2

(ii) If the system (1.1) is complete, then (2.7) implies solvability of the
approximate SEDP, thus, observability for (1.1) over ${{\mathbb{R}}_{\geq
0}}$. Indeed, let ${{t}_{0}}\in I$ and without loss of generality consider
arbitrary $s>{{t}_{0}}$. It follows by invoking the forward completeness
assumption and (2.7) that $\lim_{\nu\to\infty}{{\hat{\xi}}_{\nu}}=x(s)$, where
${{\hat{\xi}}_{\nu}}:=x(s;{{t}_{0}},{{\xi}_{\nu}},u),\nu=1,2,\ldots$ and
simultaneously (2.3) and (2.6a) hold with $s$ and $x(s)$, instead of
${{t}_{0}}$ and ${{x}_{0}}=x({{t}_{0}})$, respectively. Moreover, due to the
backward completeness, $x(s)$ is the unique vector for which
$\lim_{\nu\to\infty}{{\hat{\xi}}_{\nu}}=x(s)$.

In order to state and establish our main result, we first require the
following notations and additional assumptions for the dynamics of (1.1).
Consider ${{t}_{0}}\in I$, $\tau>{{t}_{0}}$, $u\in{{U}_{[{{t}_{0}},\tau]}}$,
$y\in{{Y}_{[{{t}_{0}},\tau],u}}$ and
$d\in{{C}^{0}}\left(\left[{{t}_{0}},\tau\right];{{\mathbb{R}}^{m}}\right)$. We
denote by $\Phi(t,{{t}_{0}})$ the fundamental matrix solution of

$\displaystyle\frac{\partial}{\partial
t}\Phi(t,{{t}_{0}})=A(t,y(t),u(t))\Phi(t,{{t}_{0}})$ (2.8a)
$\displaystyle\Phi\left({{t}_{0}},{{t}_{0}}\right)={{I}_{n\times n}}$ (2.8b)

and define the mappings:

$\Psi\left(t;{{t}_{0}},y,u\right):=\\\
\int\limits_{{{t}_{0}}}^{t}{\Phi^{\prime}\left(s,{{t}_{0}}\right)C^{\prime}\left(s,u\left(s\right)\right)C\left(s,u\left(s\right)\right)\Phi\left(s,{{t}_{0}}\right)ds},t\in\left[{{t}_{0}},\tau\right]$
(2.9)
${{\Xi}}(t;{{t}_{0}},y,d,u):=\int_{{{t}_{0}}}^{t}{{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho))C(\rho,u(\rho))\Phi(\rho,{{t}_{0}})\\\
\times\left(\int_{{{t}_{0}}}^{\rho}{\Phi}({{t}_{0}},s)f\left(s,y(s),d(s),u(s)\right)\text{d}s\right)\text{d}\rho,t\in\left[{{t}_{0}},\tau\right]$
(2.10)

We are in a position to provide our main assumptions together with the
statement and proof of our main result. A1. For system (1.1) we assume that
there exists a nonempty subset $I$ of $\mathbb{R}_{\geq 0}$ in such a way that
for all ${{t}_{0}}\in I$, $\tau>{{t}_{0}}$ close to ${{t}_{0}}$ and for each
$u\in{{U}_{[{{t}_{0}},\tau]}}$ and $y\in{{Y}_{[{{t}_{0}},\tau],u}}$ it holds:

$\displaystyle\Psi\left(t;{{t}_{0}},y(t),u(t)\right)>0,\forall
t\in({{t}_{0}},\tau]$ (2.11)

where the map $\Psi$ is given by (2.9); A2. In addition, we assume that for
every ${{t}_{0}}\in I$, $T>{{t}_{0}}$ close to ${{t}_{0}}$,
$u\in{{U}_{[{{t}_{0}},T]}}$, $y\in{{Y}_{[{{t}_{0}},T],u}}$, $\ell\in(0,1)$ and
constants $R,\theta>0$ there exists a constant
$\tau\in\left({{t}_{0}},\min\left\\{{{t}_{0}}+\theta,T\right\\}\right)$ such
that

$\|{{\Psi}^{-1}}(\cdot;{{t}_{0}},y,u)\left[{{\Xi}}(\cdot;{{t}_{0}},y,{{d}_{1}},u)-{{\Xi}}(\cdot;{{t}_{0}},y,{{d}_{2}},u)\right]{{\|}_{({{t}_{0}},\tau]}}\\\
\leq\ell\|{{d}_{1}}-{{d}_{2}}{{\|}_{[{{t}_{0}},\tau]}},\forall{{d}_{1}},{{d}_{2}}\in{{C}^{0}}\left([{{t}_{0}},\tau];{{\mathbb{R}}^{n}}\right),\\\
\text{ with }\|{{d}_{i}}{{\|}_{[{{t}_{0}},\tau]}}\leq R,i=1,2$ (2.15)

Assumption A1 is a type of persistence of excitation and A2 is a type of
contraction condition. Assumptions A1 and A2 are in general difficult to be
checked, however, they are both fulfilled for a class of nonlinear triangular
systems, under weak assumptions that are exclusively expressed in terms of
system’s dynamics (see (3.1) in the next section). We are in a position to
state and prove our main result. Our approach leads to an explicit algorithm
for the state estimation.

###### Proposition II.1

Assume that A1 and A2 are fulfilled. Then the approximate SEDP is solvable for
(1.1) over the set $I$; consequently (1.1) is observable over $I$.

###### Proof:

Let ${{t}_{0}}\in I$, $u\in{{U}_{[{{t}_{0}},\tau]}}$,
$y(\cdot)\in{{Y}_{[{{t}_{0}},\tau],u}}$, with $\tau$ as given in A1 and A2,
and let $x(\cdot)\in C^{0}\left([t_{0},\tau];\mathbb{R}^{n}\right)$ be a
solution of (1.1) corresponding to $u(\cdot)$ and $y(\cdot)$ satisfying (2.3).
Consider the trajectory ${{z}}(t)\text{:=}z(t;{{t}_{0}},{{z}}({{t}_{0}}),u)$
of the auxiliary system:

(2.16b)

for certain initial $z(t_{0})\in\mathbb{R}^{n}$. The map
${{Y}}(t)=C\left(t,u(t)\right){{z}}(t)$ is written:

$Y(t)=C\left(t,u(t)\right)\Phi(t,t_{0})z(t_{0})+C\left(t,u(t)\right)\\\
\times\int_{t_{0}}^{t}\Phi(t,s)f(s,y(s),z(s),u(s))\text{d}s$ (2.16r)

By multiplying by
${\Phi}^{\prime}(t,{{t}_{0}}){C}^{\prime}\left(t,u(t)\right)$ and integrating
we find:

$\int_{{{t}_{0}}}^{t}{{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho)){{Y}}(\rho)\text{d}\rho\\\
=(\int_{{{t}_{0}}}^{t}{{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho))C(\rho,u(\rho))\Phi(\rho,{{t}_{0}})\text{d}\rho\text{)}{{z}}({{t}_{0}})\\\
+\int_{{{t}_{0}}}^{t}({{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho))C(\rho,u(\rho))\Phi(\rho,t_{0})\\\
\times\int_{{{t}_{0}}}^{\rho}\Phi(t_{0},s)f(s,y(s),z(s),u(s))\text{d}s)\text{d}\rho$

The latter in conjunction with (2.9) - (2.11) yields:

${{z}}({{t}_{0}})={{\Psi}^{-1}}(t;{{t}_{0}},y,u)\int_{{{t}_{0}}}^{t}{{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho)){{Y}}(\rho)\text{d}\rho\\\
-{{\Psi}^{-1}}(t;{{t}_{0}},y,u){{\Xi}}(t;{{t}_{0}},y,z,u),\forall
t\in({{t}_{0}},\tau]$ (2.16y)

By considering the solution $x(\cdot)$ of (1.1) corresponding to same
$u(\cdot)$ with $y(t)=C(t,u(t))x(t),t\in[t_{0},\tau]$ and with same initial
$x(t_{0})$, it follows from (2.13) that the mappings $Y(\cdot)$ and $y(\cdot)$
coincide, therefore, from (2.14) we get:

${{x}}({{t}_{0}})={{\Psi}^{-1}}(t;{{t}_{0}},y,u)\int_{{{t}_{0}}}^{t}{{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho)){{y}}(\rho)\text{d}\rho\\\
-{{\Psi}^{-1}}(t;{{t}_{0}},y,u){{\Xi}}(t;{{t}_{0}},y,x_{\varepsilon},u),\forall
t\in({{t}_{0}},\tau]$ (2.16ab)

Let $T\in(t_{0},\tau]$ and define:

$\mathcal{F}_{T}(t;{{t}_{0}},y,z,u):=\Phi(t,{{t}_{0}}){{\Psi}^{-1}}(T;{{t}_{0}},y,u)\\\
\times\left(\int_{{{t}_{0}}}^{T}{{{\Phi}^{\prime}}}(\rho,{{t}_{0}}){C}^{\prime}(\rho,u(\rho))y(\rho)\text{d}\rho-\Xi(T;{{t}_{0}},y,z,u)\right)\\\
+\int_{{{t}_{0}}}^{t}{\Phi}(t,\rho)f(\rho,y(\rho),z(\rho),u(\rho))\text{d}\rho,\\\
t\in[{{t}_{0}},T],z(\cdot)\in{{C}^{0}}\left([{{t}_{0}},T];{{\mathbb{R}}^{n}}\right)$
(2.16ag)

Then, by (2.15) and (2.16) we have:

${{\mathcal{F}}_{T}}(t;{{t}_{0}},y,{{x}},u)={{x}}(t),\forall
t\in[{{t}_{0}},T]$ (2.16ah)

Next, consider a strictly increasing sequence
$\left\\{R_{\nu}\in\mathbb{R}_{>0},\nu=1,2,\ldots\right\\}$ defined as:

$R_{\nu+1}=2R_{\nu},\nu=1,2,\ldots,\text{ wth }R_{1}=1$ (2.16ai)

Since, due to continuity of $x(\cdot)$, the set
$\left\\{x(t),t\in[t_{0},\tau]\right\\}$ is bounded, there exists an integer
$k\geq 1$ such that

$\|x(\cdot)\|_{[t_{0},\tau]}<R_{k}$ (2.16aj)

Let

$\ell\in(0,1/2]$ (2.16ak)

By virtue of (2.1), (2.12) and (2.16) it follows that for the above $\ell$
there exists a decreasing continuous function
$T=T(R):\mathbb{R}_{>0}\to(t_{0},\tau]$ with $\lim_{R\to+\infty}T(R)=t_{0}$
and such that

$\|\mathcal{F}_{T}(\cdot;{{t}_{0}},y,{{d}_{1}},u)-\mathcal{F}_{T}(\cdot;{t_{0}},y,{{d}_{2}},u)\|_{(t_{0},T]}\leq\ell\|d_{1}-d_{2}\|_{[{{t}_{0}},T]},\\\
T:=T(R),\forall d_{1},d_{2}\in
C^{0}\left([t_{0},T(R)];\mathbb{R}^{n}\right),\text{ for which }\\\
\max\left\\{\|d_{i}\|_{[t_{0},T(R)]},i=1,2\right\\}\leq R$ (2.16ao)

Finally, define:

$\displaystyle t_{\nu}:=T(R_{\nu});$ (2.16apa) $\displaystyle\begin{aligned}
z_{\nu+1}(t):=\mathcal{F}_{t_{\nu}}(t;{{t}_{0}},y,{{z}_{\nu}},u);\\\
t\in[{{t}_{0}},{{t}_{\nu}}],\nu=k,k+1,k+2,\ldots\end{aligned}$ (2.16apb)

with arbitrary

$z_{k}\in
C^{0}\left([t_{0},t_{k}];\mathbb{R}^{n}\right):\|z_{k}\|_{[t_{0},t_{k}]}<R_{k}$
(2.16aq)

Then by (2.19) and (2.21)-(2.23) we get

$\|\mathcal{F}_{t_{k}}(\cdot;{{t}_{0}},y,z_{k},u)-\mathcal{F}_{t_{k}}(\cdot;{t_{0}},y,x,u)\|_{[t_{0},t_{k}]}\leq\ell\|z_{k}-x\|_{[{{t}_{0}},t_{k}]}$
(2.16ar)

According to (2.22b) let
${{z}_{k+1}}(\cdot):={{\mathcal{F}}_{t_{k}}}(\cdot;{{t}_{0}},y,{{z}_{k}},u)$.
Then, from (2.17), (2.21)-(2.24) and the fact that the sequence
$\left\\{{{t}_{\nu}}\in({{t}_{0}},\tau]\right\\}$ is decreasing, we have:

$\|{{z}_{k+1}}-x{{\|}_{[{{t}_{0}},{{t}_{k+1}}]}}=\|{{\mathcal{F}}_{{{{t}_{k}}}}}(\cdot;{{t}_{0}},y,{{z}_{k}},u)-x{{\|}_{[{{t}_{0}},{{t}_{k+1}}]}}\\\
=\|{{\mathcal{F}}_{{{{t}_{k}}}}}(\cdot;{{t}_{0}},y,{{z}_{k}},u)-{{\mathcal{F}}_{{{t}_{k}}}}(\cdot;{{t}_{0}},y,x,u){{\|}_{[{{t}_{0}},{{t}_{k+1}}]}}\\\
\leq\|{{\mathcal{F}}_{{{{t}_{k}}}}}(\cdot;{{t}_{0}},y,{{z}_{k}},u)-{{\mathcal{F}}_{{{t}_{k}}}}(\cdot;{{t}_{0}},y,x,u){{\|}_{[{{t}_{0}},{{t}_{k}}]}}\leq\ell\|{{z}_{k}}-x{{\|}_{[{{t}_{0}},{{t}_{k}}]}}$
(2.16av)

therefore, by invoking (2.18), (2.19), (2.20), (2.23) and (2.25) it follows:

$\|{{z}_{k+1}}{{\|}_{[{{t}_{0}},{{t}_{k+1}}]}}\leq\|x{{\|}_{[{{t}_{0}},{{t}_{k+1}}]}}+\ell\|{{z}_{k}}-x{{\|}_{[{{t}_{0}},{{t}_{k}}]}}<{{R}_{k}}+2\ell{{R}_{k}}\\\
\leq{{R}_{k+1}}$ (2.16ax)

Quite similarly, by induction we get:

$\|{{z}_{\nu+1}}-x{{\|}_{[{{t}_{0}},{{t}_{\nu+1}}]}}\leq{{\ell}^{\nu+1-k}}\|{{z}_{k}}-x{{\|}_{[{{t}_{0}},{{t}_{1}}]}};\\\
\quad\|{{z}_{\nu}}{{\|}_{[{{t}_{0}},{{t}_{\nu+1}}]}}\leq{{R}_{\nu}},\forall\nu=k,k+1,k+2,\ldots$
(2.16ba)

which implies

${{\xi}_{\nu}}:={{z}_{\nu}}({{t}_{\nu}})\to
x({{t}_{0}});{{X}_{\nu}}({{t}_{0}},y,u):={{\xi}_{\nu}}$ (2.16bb)

where the values ${{\xi}_{\nu}}$ above are exclusively dependent on the values
of the input $u$ and the output $y$ and the dynamics of system and they are
independent of any time-derivatives of $u$ and $y$, thus both (2.6a) and
(2.6b) are fulfilled. We conclude that the approximate SEDP is solvable for
(1.1) over $I$, therefore, system (1.1) is observable over $I$. ∎ The
existence result of Proposition 2.1 does not in general determine explicitly
the desired sequence of mappings ${{X}_{\nu}}$ exhibiting (2.27). The reason
is that, although existence of the constant $k$ satisfying (2.19) is
guaranteed from boundedness of
$\left\\{x(t),t{{\in}{[{{t}_{0}},\tau]}}\right\\}$, its precise determination
requires knowledge of a bound of the previous set, which, in general, is not
available. The rest part of this section is devoted for the establishment of a
constructive algorithm, exhibiting the state determination. The corresponding
procedure is based on the approach given in proof of Proposition 2.1 plus some
appropriate modifications.

Algorithm

To simplify the procedure, we distinguish two cases:

Case I: First, we assume that a bounded region of the state space is a priori
known, where the unknown initial state of (1.1) belongs. Particularly, assume
that for all ${{t}_{0}}\in I$, almost all $\tau>{{t}_{0}}$ near ${{t}_{0}}$
and input $u\in{{U}_{[{{t}_{0}},\tau]}}$, an open ball ${{B}_{R}}$ of radius
$R>0$ centered at zero is known, such that the corresponding set of outputs of
(1.1) is modified as follows:

${{Y}_{[{{t}_{0}},\tau],u}}:=\\{y\in{{C}^{0}}\left([{{t}_{0}},\tau];{{\mathbb{R}}^{k}}\right):y(t)=C\left(t,u(t)\right)\\\
\times x\left(t;{{t}_{0}},{{x}_{0}},u\right),\text{ a.e.}\
t\in[{{t}_{0}},\tau],\text{ for certain }{{x}_{0}}\in{{B}_{R}}\\}$ (2.16be)

For the case above we adopt a slight modification of the approach used for the
proof of Proposition 2.1. Our proposed algorithm contains two steps: Step 1:
Define

$R_{1}:=R$ (2.16bf)

where the latter is involved in (2.28). Notice that, due to the additional
assumption (2.28), it follows that (2.19) holds with $k=1$ and for $\tau$
close to $t_{0}$. Next, consider a strictly increasing sequence
$\left\\{{{R}_{\nu}}\in{{\mathbb{R}}_{>0}},\nu=1,2,\ldots\right\\}$ satisfying
the first equality of (2.18), namely,

$R_{\nu+1}=2R_{\nu},\nu=1,2,\ldots$ (2.16bg)

and with ${{R}_{1}}$ as above. We set $\ell=1/2$ and find a decreasing
sequence ${{t}_{\nu}}\in{{\mathbb{R}}_{>0}},\nu=1,2,\ldots$ with
${{t}_{\nu}}\to{{t}_{0}}$ and in such a way that

$\|{{\mathcal{F}}_{{{t}_{\nu}}}}(\cdot;{{t}_{0}},y,{{d}_{1}},u)-{{\mathcal{F}}_{{{t}_{\nu}}}}(\cdot;{{t}_{0}},y,{{d}_{2}},u){{\|}_{[{{t}_{0}},{{t}_{\nu}}]}}\leq\ell\|{{d}_{1}}-{{d}_{2}}{{\|}_{[{{t}_{0}},{{t}_{\nu}}]}},\\\
\quad\forall{{d}_{1}},{{d}_{2}}\in{{C}^{0}}\left([{{t}_{0}},{{t}_{\nu}}];{{\mathbb{R}}^{n}}\right):\\\
\max\left\\{\|{{d}_{i}}{{\|}_{[{{t}_{0}},{{t}_{\nu}}]}},i=1,2\right\\}\leq{{R}_{{}_{\nu}}},\nu=1,2,\ldots$
(2.16bk)

Step 2: Consider the sequence
${{z}_{\nu+1}}(t):={{\mathcal{F}}_{{{{t}_{\nu}}}}}(t;{{t}_{0}},y,{{z}_{\nu}},u),t\in[{{t}_{0}},t_{\nu}]$
with arbitrary initial
${{z}_{1}}(\cdot)\in{{C}^{0}}\left([{{t}_{0}},{{t}_{k}}];{{\mathbb{R}}^{n}}\right)$
satisfying (2.23) with $k=1$ and set

${{X}_{\nu}}({{t}_{0}},y,u):={{z}_{\nu}}({{t}_{\nu}}),\nu=1,2,\ldots$ (2.16bl)

It then follows that (2.27) holds with unique $x({{t}_{0}})\in{{B}_{R}}$
satisfying (2.3). Particularly, we have:

$\|{{z}_{\nu+1}}-x{{\|}_{[{{t}_{0}},{{t}_{\nu+1}}]}}\leq{{\ell}^{\nu}}\|{{z}_{1}}-x{{\|}_{[{{t}_{0}},{{t}_{1}}]}},\nu=1,2,\ldots$
(2.16bm)

therefore, the sequence of mappings ${{X}_{\nu}}$, as defined by (2.32),
exhibits the state determination. above satisfies the desired (2.5) and (2.6).
Case II (General Case): We now provide an algorithm, which exhibits the state
determination for the general case, without any additional assumption. The
algorithm contains two steps: Step 1: Repeat the same procedure followed in
Case I, with $R=1,2,3,\ldots$ and construct a sequences of mappings

$\displaystyle\begin{aligned}
z_{\nu+1}^{i}(t):=\mathcal{F}_{t_{\nu}^{i}}(t;{{t}_{0}},y,{{z}_{\nu}},u),t\in[{{t}_{0}},t_{\nu}^{i}],\\\
z_{1}^{i}(\cdot):=0,\nu=1,2,3,\ldots;i=1,2,3,\ldots\end{aligned}$ (2.16bna)
associated with appropriate decreasing sequences
$\left\\{t_{\nu}^{i}\in(t_{0},\tau]\right\\},i=1,2,3,\ldots$, with
$\lim_{\nu\to\infty}t_{\nu}^{i}=t_{0}$, by pretending that
$\|x(\cdot){{\|}_{[{{t}_{0}},\tau]}}<i,i=1,2,\ldots$ and in such a way that,
if we define $\xi_{\nu}^{i}:=z_{\nu}^{i}(t_{\nu}^{i})$, we have:
$\displaystyle\begin{aligned}
\left|\xi_{\nu+1}^{i}-x({{t}_{0}})\right|\leq{{\ell}^{\nu}}\|z_{1}^{i}-x{{\|}_{[{{t}_{0}},t_{1}^{i}]}};\|z_{1}^{i}\|_{[t_{0},t_{1}^{i}]}<i,\\\
\forall\nu,i=1,2,\ldots,\text{ provided that
}\|x(\cdot){{\|}_{[{{t}_{0}},\tau]}}<i\end{aligned}$ (2.16bnb)

Step 2: Define

$X_{\nu}({{t}_{0}},y,u):=\xi_{\nu}^{\nu},\nu=1,2,\ldots$ (2.16bo)

Notice that, since the set $\left\\{x(t),t{{\in}{[{{t}_{0}},\tau]}}\right\\}$
is bounded, there exists an integer $k$ such that
$\|x(\cdot){{\|}_{[{{t}_{0}},\tau]}}<k<k+1<k+2<\ldots$. The latter in
conjunction with (2.33) yields:

$\left|\xi_{\nu}^{\nu}-x({{t}_{0}})\right|\leq{{\ell}^{\nu-1}}\|z_{1}^{\nu}-x{{\|}_{[{{t}_{0}},t^{\nu}_{1}]}},\leq\ell^{\nu-1}(\nu+k),\nu=k+1,k+2,\ldots$
(2.16bp)

which, due to selection $\ell=1/2$, implies
${{X}_{\nu}}({{t}_{0}},y,u):=\xi_{\nu}^{\nu}\to x({{t}_{0}})$. We conclude
that for the general case the sequence of mappings ${{X}_{\nu}}$, as defined
by (2.34), exhibits the state determination. Finally, it should be noted that,
according to the methodology above, contrary to the approach adopted in the
proof of Proposition 2.1, the specific knowledge of $k$ satisfying (2.35) is
not required.

##  III Application

In this section we apply the results of Section II to triangular systems (1.1)
of the form:

$\displaystyle\begin{aligned}
\dot{x}_{i}=a_{i+1}(t,x_{1},u)x_{i+1}+f_{i}(t,x_{1},u),i=1,\ldots,n-1,\\\
\dot{x}_{n}=f_{n}(t,x_{1},\ldots,x_{n},u)\end{aligned}$ (2.16aa)
$\displaystyle(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n},u\in\mathbb{R}^{m},$
with output $\displaystyle y=x_{1}$ (2.16ab)

where we make the following assumptions: H1 (Regularity Assumptions). It is
assumed that for each
$(x,y,u)\in{{\mathbb{R}}^{n}}\times\mathbb{R}\times{{\mathbb{R}}^{m}}$ the
mappings ${{a}_{i}}(\cdot,y,u),\,\,i=2,\ldots,n$,
${{f}_{i}}(\cdot,{{x}_{1}},u)$, $i=1,\ldots,n-1$ and
${{f}_{n}}(\cdot,{{x}_{1}},\ldots,{{x}_{n}},u)$ are measurable and locally
essentially bounded and for each fixed $t\geq 0$ and $u\in{{\mathbb{R}}^{m}}$
the mappings
${{a}_{i}}(t,\cdot,u),i=2,\ldots,n,{{f}_{i}}(t,\cdot,u),i=1,\ldots,n-1$ and
${{f}_{n}}(t,\ldots,u)$ are (locally) Lipschitz.

Obviously, (3.1) has the form of (1.1) with

$\displaystyle A(t,y,u):=$
$\displaystyle{\begin{bmatrix}0&a_{2}(t,y,u)&0&\cdots&0\\\
0&0&a_{3}(t,y,u)&\cdots&0\\\ \vdots&\vdots&\ddots&\\\
0&0&\cdots&&a_{n}(t,y,u)\\\ 0&0&0&\cdots&0\end{bmatrix}}$ (2.16ba)
$\displaystyle
f:=[\begin{matrix}{{f}_{1}}&{{f}_{2}}&\cdots&{{f}_{n}}\end{matrix}]^{\prime}$
(2.16bb) $\displaystyle
C\left(t,u\right):=C=\left[\begin{matrix}1&0&\cdots&0\end{matrix}\right]$
(2.16bc)

We also make the following observability assumption: H2. There exists a
measurable set $I\subset{{\mathbb{R}}_{\geq 0}}$ with nonempty interior such
that for all ${{t}_{0}}\in I$, $\tau>{{t}_{0}}$ close to ${{t}_{0}}$ and for
each
$u\in{{U}_{[{{t}_{0}},\tau]}}:={{L}_{\infty}}([{{t}_{0}},\tau];{{\mathbb{R}}^{m}})$
and $y\in Y_{[t_{0},\tau],u}$ it holds:

$\displaystyle\prod\limits_{i=2}^{n}{{{a}_{i}}}({{t}_{0}})\neq
0;{{a}_{i}}({{t}_{0}})\text{
:=}{{a}_{i}}({{t}_{0}},y({{t}_{0}}),u({{t}_{0}}))$ (2.16c)

###### Proposition III.1

For the system (3.1) assume that H1 and H2 hold with
${{U}_{[{{t}_{0}},\tau]}}:={{L}_{\infty}}([{{t}_{0}},\tau];{{\mathbb{R}}^{m}})$
for certain $\tau>{{t}_{0}}$ close to ${{t}_{0}}\in I$. Then there exists a
set $\hat{I}\subset I$ with $\text{cl}\hat{I}=I$ such that the approximate
SEDP is solvable over $\hat{I}$ for the system (3.1) by employing the
methodology of Proposition 2.1; consequently, (3.1) is observable over
$\hat{I}$.

###### Remark III.1

A stronger version of assumption (3.3), is required in [8], [9], [25], [27],
[28], for the construction of Luenberger type observers for a more general
class of triangular systems. Particularly, in all previously mentioned works
it is further imposed that the mappings $a_{i}(\cdot,\cdot,\cdot)$ are
$C^{1}$. We note that the second conclusion of Proposition 3.1 concerning
observability can alternatively be obtained under H1 and H2 as follows: By
exploiting (3.3) and applying successive differentiation with respect to time,
we can determine a map $X$ satisfying (2.4) with the information of the time-
derivatives of $u$ and output $y$ (details are left to the reader). But this
map is not acceptable for the solvability of SEDP for (3.1), due to the
additional requirements of Definition 2.2 that the candidate $X$ should be
independent of the time-derivatives of $u$ and $y$.

###### Proof:

We establish that the assumptions H1 and H2 guarantee that conditions A1 and
A2 of previous section are fulfilled for (3.1), therefore by invoking
Proposition 2.1 we get the desired statement. We first evaluate the
fundamental solution $\Phi(t,{{t}_{0}})$ of (2.8) with $A(t,y,u)$ as given by
(3.2a) for certain $t_{0}\in I$. We find:

$\Phi(t,t_{0})=\\\
{\begin{bmatrix}\varepsilon_{11}(=1)&\varepsilon_{12}(t)(t-t_{0})&\cdots&\varepsilon_{1n}(t)(t-t_{0})^{n-1}\\\
0&\varepsilon_{21}(=1)&\cdots&\varepsilon_{2,n-1}(t)(t-t_{0})^{n-2}\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&\varepsilon_{n1}(=1)\end{bmatrix}}$
(2.16f)

where each function
${{\varepsilon}_{ij}}:[{{t}_{0}},\tau]\to\mathbb{R},i,j=1,2,\ldots,n$
satisfies

$\varepsilon_{ij}(t)=E_{ij}\left(1+Z_{ij}(t)\right)$ (2.16ga) for certain
constants ${{E}_{ij}}\in\mathbb{R}$ and functions
${{Z}_{ij}}\in{{L}_{\infty}}([{{t}_{0}},\tau];\mathbb{R})$ with
$\underset{t\to{{t}_{0}}}{\mathop{\lim}}\,{{Z}_{ij}}(t)=0$ (2.16gb)

Particularly, due to (3.3), we have: for certain nonzero constants
${{c}_{i}}$. Notice, that the above representation is feasible for almost all
$t_{0}\in I$ due to our regularity assumptions concerning ${{a}_{i}}$. For
simplicity, we may assume next that (3.4) - (3.6) hold for every $t_{0}\in I$.
We now calculate by taking into account (3.2c) and (3.4):

$\displaystyle
C\Phi\left(t,{{t}_{0}}\right)=[{{\varepsilon}_{11}}(t),{{\varepsilon}_{12}}(t)(t-{{t}_{0}}),\cdots,{{\varepsilon}_{1n}}(t){{(t-{{t}_{0}})}^{n-1}}]$
(2.16i)

Notice that $\Psi$, as defined by (2.9), satisfies (2.11) since otherwise,
there would exist sequences
$\upsilon^{i}=(\upsilon_{1}^{i},\upsilon_{2}^{i},\ldots,\upsilon_{n}^{i})\in\mathbb{R}^{n}\setminus\\{0\\},{{{t}_{i}}\in({{t}_{0}},\tau]}$
and a nonzero vector
$\upsilon=[{{\upsilon}_{1}},{{\upsilon}_{2}},\cdots,{{\upsilon}_{n}}]\in{{\mathbb{R}}^{n}}$
with
$\lim_{i\to\infty}\upsilon^{i}=\upsilon,\lim_{i\to\infty}{{t}_{i}}={{t}_{0}}$
and in such a way that
$C\Phi({{t}_{i}},{{t}_{0}})\upsilon^{i}=0,i=1,2,\ldots$. Then by using (3.7)
we get
${{\upsilon}_{1}^{i}}{{\varepsilon}_{11}}({{t}_{i}})+{{\upsilon}_{2}^{i}}{{\varepsilon}_{12}}({{t}_{i}})({{t}_{i}}-{{t}_{0}})+\ldots+{{\upsilon}_{n}^{i}}{{\varepsilon}_{1n}}({{t}_{i}}){{({{t}_{i}}-{{t}_{0}})}^{n-1}}=0$,
which by virtue of (3.5) and (3.6) implies that $\upsilon=0$, a contradiction.
We conclude that relation (2.11) of A1 holds with
${{U}_{[{{t}_{0}},\tau]}}={{L}_{\infty}}([{{t}_{0}},\tau];{{\mathbb{R}}^{m}})$.
In order to establish A2, we calculate, according to definition (2.9) and by
using (3.7):

$\Psi={\begin{bmatrix}\varepsilon_{11}(t)\Delta t&\varepsilon_{12}(t)(\Delta
t)^{2}&\cdots&\varepsilon_{1n}(t)(\Delta t)^{n}\\\ \varepsilon_{21}(t)(\Delta
t)^{2}&\varepsilon_{22}(t)(\Delta t)^{3}&\cdots&\varepsilon_{2n}(t)(\Delta
t)^{n+1}\\\ \vdots&\vdots&&\vdots\\\ \varepsilon_{n1}(t)(\Delta
t)^{n}&\varepsilon_{n2}(t)(\Delta t)^{n+1}&\cdots&\varepsilon_{nn}(t)(\Delta
t)^{2n-1}\end{bmatrix}}$ (2.16j)

where $\Delta t:=t-t_{0}$ and the functions
${{\varepsilon}_{ij}}:[{{t}_{0}},\tau]\to\mathbb{R}$ above satisfy (3.5).
Define ${{d}_{1}}=({{x}_{2}},\ldots,{{x}_{n}}{)}^{\prime}$,
${{d}_{2}}=({{\bar{x}}_{2}},\ldots,{{\bar{x}}_{n}}{)}^{\prime}$ and let
$f:=[f_{1},f_{2},\cdots,f_{n-1},f_{n}]^{\prime}$ and

$\displaystyle\Delta
f(\cdot,{{d}_{1}},{{d}_{2}},u):=f(\cdot,{{d}_{2}},u)-f(\cdot,{{d}_{1}},u)$
(2.16k)

By (3.1a), (3.2c), (3.4) and (3.9) we find:

$\int_{{{t}_{0}}}^{\rho}{\Phi}(t_{0},s)\Delta
f(s,{{d}_{1}}(s),{{d}_{2}}(s),u(s))\text{d}s=\\\
\left[\varepsilon_{1}(\rho)(\Delta\rho)^{n},\varepsilon_{2}(\rho)(\Delta\rho)^{n-1},\cdots,\varepsilon_{n-1}(\rho)(\Delta\rho)^{2},\varepsilon_{n}(\rho)\Delta\rho\right]^{\prime}$
(2.16n)

$\rho\geq{{t}_{0}}$ near ${{t}_{0}}$, $\Delta\rho:=t_{0}-\rho$, where the
functions
${{\varepsilon}_{i}}(\cdot)={{\varepsilon}_{i}}(\cdot;{{d}_{1}}(\cdot),{{d}_{2}}(\cdot))$
have the form:

$\displaystyle{{\varepsilon}_{i}}(t)={{E}_{i}}(1+{{Z}_{i}}(t)),\
li{{m}_{t\to{{t}_{0}}}}{{Z}_{i}}(t)=0,i=1,\ldots,n$ (2.16o)

for certain ${{E}_{i}}\in\mathbb{R},$
${{Z}_{i}}\in{{L}_{\infty}}([{{t}_{0}},\tau];\mathbb{R})$, and in such a way
that, due to (3.9) and Lipschitz continuity property of ${{f}_{n}}$, the
following holds for every $R>0$:

$|\varepsilon_{i}(t)|\leq C\|d_{1}-d_{2}\|_{[t_{0},\tau]},\forall
t\in[t_{0},\tau],\tau\text{ near }t_{0},\\\
{{d}_{1}},{{d}_{2}}\in{{C}^{0}}\left([{{t}_{0}},\tau];{{\mathbb{R}}^{n}}\right)\text{
with }\|{{d}_{i}}{{\|}_{[{{t}_{0}},\tau]}}\leq R,i=1,2$ (2.16r)

for certain constant $C>0$. Also, we evaluate from (3.8):

$\Psi^{-1}=\\\ \begin{bmatrix}\varepsilon_{11}(t)(\Delta
t)^{-1}&\varepsilon_{12}(t)(\Delta t)^{-2}&\cdots&\varepsilon_{1n}(t)(\Delta
t)^{-n}\\\ \varepsilon_{21}(t)(\Delta t)^{-2}&\varepsilon_{22}(t)(\Delta
t)^{-3}&\cdots&\varepsilon_{2n}(t)(\Delta t)^{-n-1}\\\
\vdots&\vdots&&\vdots\\\ \varepsilon_{n1}(t)(\Delta
t)^{-n}&\varepsilon_{n2}(t)(\Delta t)^{-n-1}&\cdots&\varepsilon_{nn}(t)(\Delta
t)^{-2n+1}\end{bmatrix}$ (2.16u)

where $\Delta t:=t-t_{0}$ and ${{\varepsilon}_{ij}}$ above satisfy (3.5a).
From (2.10), (3.7) and (3.10) - (3.12) we also find:

$\Xi(t;{{t}_{0}},y,{{d}_{1}},u)-\Xi(t;{{t}_{0}},y,{{d}_{2}},u)=\\\
{{\left[{{\varepsilon}_{1}}(t){{(\Delta
t)}^{n+1}},{{\varepsilon}_{2}}(t){{(\Delta
t)}^{n+2}},\cdots,{{\varepsilon}_{n}}(t){{(\Delta t)}^{2n}}\right]}^{\prime}}$
(2.16x)

for $t$ near ${{t}_{0}}$, where $\Delta t:=t-t_{0}$ and each
${{\varepsilon}_{i}},i=1,\ldots,n$ above satisfy again (3.11) and (3.12). The
latter in conjunction with (3.9), (3.13) and (3.14) implies A2. To be precise,
the following holds: For every ${{t}_{0}}\in I$, $T>{{t}_{0}}$ close to
${{t}_{0}}$, $u\in{{U}_{[{{t}_{0}},T]}}$, $y\in{{Y}_{[{{t}_{0}},T],u}}$ and
constants $\ell\in(0,1)$ and $R,\theta>0$, a constant
$\tau\in({{t}_{0}},\min\left\\{{{t}_{0}}+\theta,T\right\\})$ can be found
satisfying (2.12). We conclude that both A1 and A2 are fulfilled for the case
(3.1), hence, according to Proposition 2.1, the approximate SEDP is solvable
for (3.1) over a set $\hat{I}\subset I$ with $\text{cl}\hat{I}=I$. ∎

###### Example III.1

We illustrate the nature of our methodology by considering the elementary case
of the planar single-input triangular system
${{{\dot{x}}}_{1}}={{x}_{2}}u,{{{\dot{x}}}_{2}}={{x}_{1}}-x_{2}^{3}$ with
output $y={{x}_{1}}$ that has the form (3.1) with

$A:=\left[\begin{matrix}0&u\\\ 0&0\\\ \end{matrix}\right]\text{and
}f:=\left(0,{{x}_{1}}-x_{2}^{3}\right)^{\prime}.$

We may assume that each admissible input $u$ is any nonzero measurable and
essentially locally bounded function and for simplicity, let $u(t)=1$ for $t$
near zero. Obviously, the system above satisfies H1 and H2. Let us choose
$({{x}_{1}}(0),{{x}_{2}}(0))=(2,\,0)$ as initial condition and calculate the
corresponding output trajectory $y={{x}_{1}}$ (see Figure 1 below).

Figure 1: Output of the System

We next apply the methodology suggested in the previous section, in order to
confirm that our proposed algorithm converges to
$x(0)=({{x}_{1}}(0),{{x}_{2}}(0))$ above. For simplicity, let us assume that
is a priori known that the “unknown” initial state $x(0)$ is contained into
the ball ${{B}_{R}}$ of radius $R=3$ centered at zero. We take
${{R}_{1}}=3,{{R}_{2}}=6,{{R}_{3}}=12,{{R}_{4}}=24,{{R}_{5}}=48,\ldots$ and
$\ell=0.5$ as in the proposed algorithm (Case I). By taking into account the
known values of $y(\cdot)$, we find a decreasing sequence
$\left\\{{{t}_{\nu}}\right\\}$ satisfying (2.31) converging to $t_{0}=0$;
particularly, take ${{t}_{1}}=5\times{{10}^{-4}}$
${{t}_{2}}=1.3\times{{10}^{-4}},\ {{t}_{3}}=3.2\times{{10}^{-5}},\
{{t}_{4}}=7.8\times{{10}^{-6}},\ {{t}_{5}}=1.9\times{{10}^{-6}},\ldots$. Then,
choose an arbitrary constant initial map, say
${{z}_{1}}(\cdot)=(z_{1}^{1}(\cdot),z_{1}^{2}(\cdot))$; with
$z_{1}^{1}\left(t\right)=0$ and $z_{1}^{2}\left(t\right)=1$, for
$t\in[{{t}_{0}}=0,{{t}_{1}}=5\times{{10}^{-4}}]$ and successively apply
(2.22b), in order to evaluate the desired sequence
$\left\\{{{z}_{\nu}}({{t}_{\nu}}):=(z_{\nu}^{1}({{t}_{\nu}}),z_{\nu}^{2}({{t}_{\nu}})\right\\},\nu=1,2,\ldots$.
Figures 2 and 3 present the corresponding values of errors
$e_{\nu}^{1}({{t}_{\nu}}):=z_{\nu}^{1}({{t}_{\nu}})-{{x}_{1}}(0),e_{\nu}^{2}({{t}_{\nu}}):=z_{\nu}^{2}({{t}_{\nu}})-{{x}_{2}}(0),\nu=1,2,\ldots$
and confirms that the evaluated pair of terms
${{z}_{\nu}}\left({{t}_{\nu}}\right)$ converges to the pair
$({{x}_{1}}(0),{{x}_{2}}(0))=(2,0)$.

Figure 2: Error $e_{\nu}^{1}$ Figure 3: Error $e_{\nu}^{2}$

Finally, we remark that, since the system is forward complete, then for any
$T>{{t}_{0}}$ the sequence of mappings
${{X}_{\nu}}(0,y,u):=x(t;0,{{\xi}_{\nu}},u),t\in[0,T],\nu=1,2,\ldots$ with
${{\xi}_{\nu}}:={{z}_{\nu}}\left({{t}_{\nu}}\right)$ uniformly approximates
the unknown solution $x(\cdot;0,x(0),u)$ on the interval $[{{t}_{0}},T]$.

##  IV Additional Hypotheses and Hybrid Observer

In this section we briefly present a hybrid-observer technique for the state
estimation for (1.1). The proof of the following proposition is based on a
modification of the approach employed in Section II.

###### Proposition IV.1

For the system (1.1) we make the same assumptions with those imposed in
statement of Proposition 2.1. Also, assume that for any ${{t}_{0}}\in I$ and
input $u\in{{U}_{[{{t}_{0}},+\infty)}}$ there exists a constant $C>0$ such
that

$\displaystyle\left|f\left(t,y,{{z}_{1}},u(t)\right)-f\left(t,y,{{z}_{2}},u(t)\right)\right|\leq
C\left|{{z}_{1}}-{{z}_{2}}\right|,$
$\displaystyle\forall(t,y)\in[t_{0},\infty)\times\mathbb{R},{{z}_{1}},{{z}_{2}}\in\mathbb{R}^{n}$
(2.16a)

Then, there exists a sequence
$\xi_{\nu}=\xi_{\nu}(t_{0},y,u)\in\mathbb{R}^{n},\nu=0,1,2,\ldots$ such that,
if for any arbitrary constant $h>t_{0}$ we define:

(2.16bb)

where $\sigma:=h-t_{0}$, then the system below exhibits the global state
estimation of (1.1):

$\dot{\hat{x}}\left(t\right)=A(t,y,u)\hat{x}+f(t,y,\hat{x},u),\
t\in\left[{{t}_{0}}+\nu\sigma,{{t}_{0}}+(\nu+1)\sigma\right)$ (2.16bca)
$\hat{x}\left({{t}_{0}}+\nu\sigma\right)={{m}_{\nu}},\ \nu=0,1,2,\ldots$
(2.16bcb)

particularly, it holds:

###### Proof:

Let $x(\cdot)=x(\cdot;{{t}_{0}},{{x}_{0}},u),{{t}_{0}}\in I$ be a solution of
(1.1) corresponding to $u(\cdot)$. Let $h>{{t}_{0}}$ and consider the sequence
${{C}_{\nu}}:={{(\exp hC)}^{\nu+1}},\nu=1,2,\ldots$, with $C$ as defined in
(4.1), and let
$\left\\{{{\ell}_{\nu}}\right\\},{{\ell}_{\nu}}\in(0,1/2],\nu=1,2,\ldots$ be a
decreasing sequence with

$\displaystyle\lim_{\nu\to\infty}\ell_{\nu-1}C_{\nu}=0$ (2.16be)

We next proceed by using a generalization of the procedure employed for the
proof of Proposition 2.1. First, we find a decreasing sequence
${{t}_{\nu}}\in({{t}_{0}},h]$, $,\nu=0,1,2,\ldots$ with
${{t}_{\nu}}\to{{t}_{0}}$ such that

$\|{{\mathcal{F}}_{{{t}_{\nu}}}}(\cdot;{{t}_{0}},y,{{d}_{1}},u)-{{\mathcal{F}}_{{{t}_{\nu}}}}(\cdot;{{t}_{0}},y,{{d}_{2}},u){{\|}_{[{{t}_{0}},{{t}_{\nu}}]}}\leq{{\ell}_{\nu}}\|{{d}_{1}}-{{d}_{2}}{{\|}_{[{{t}_{0}},{{t}_{\nu}}]}},\\\
\forall{{d}_{1}},{{d}_{2}}\in{{C}^{0}}\left([{{t}_{0}},{{t}_{\nu}}];{{\mathbb{R}}^{n}}\right):\max\left\\{\|{{d}_{i}}{{\|}_{[{{t}_{0}},{{t}_{\nu}}]}},i=1,2\right\\}\leq{{R}_{\nu}},\\\
\nu=k,k+1,k+2,\ldots$ (2.16bi)

with $\left\\{{{R}_{\nu}}\right\\}$ and $k$ as defined by (2.18) and (2.19),
respectively. Then, consider the sequence of mappings
${{z}_{\nu}}(\cdot)\in{{\mathbb{R}}^{n}},\nu=k,k+1,k+2,\ldots$, as precisely
defined in (2.22) and again define:

${{\xi}_{\nu}}:={{z}_{\nu}}({{t}_{\nu}}),\nu=k,k+1,\ldots$ (2.16bj)

Then, as in proof of Propostion 2.1 we can show, by exploiting (4.6), that

$\left|{{\xi}_{\nu}}-{{x}_{0}}\right|\leq{{\ell}_{\nu}}{{\ell}_{\nu-1}}\cdots{{\ell}_{k}}\left|{{\xi}_{k}}-{{x}_{0}}\right|,\forall\nu=k+1,k+2,\ldots$
(2.16bk)

We are now in a position to show (4.4). We take into account (4.1)-(4.3),
(4.5), (4.8), definition of ${{C}_{\nu}}$ and consider the difference between
the integral representation of the solutions of (1.1a) and (4.3a). Then, by
successively applying the Gronwall - Bellman inequality, we can estimate:

${{\left\|\hat{x}(\cdot;{{t}_{0}}+\nu\sigma,{{m}_{\nu}},u)-x(\cdot;{{t}_{0}},{{x}_{0}},u)\right\|}_{[{{t}_{0}}+\nu\sigma,{{t}_{0}}+(\nu+1)\sigma]}}\\\
\leq{{\ell}_{k}}{{\ell}_{k+1}}\cdots{{\ell}_{\nu-1}}{{C}_{\nu}}\left|{{\xi}_{0}}-{{x}_{0}}\right|\leq{{\ell}_{\nu-1}}{{C}_{\nu}}\left|{{\xi}_{0}}-{{x}_{0}}\right|,\\\
\nu=k+1,k+2,\ldots$ (2.16bo)

and the above, in conjunction with (4.5), asserts that

$\underset{\nu\to\infty}{\mathop{\lim}}\,{{\left\|\hat{x}(\cdot;{{t}_{0}}+\nu\sigma,{{m}_{\nu}},u)-x(\cdot;{{t}_{0}},{{x}_{0}},u)\right\|}_{[{{t}_{0}}+\nu\sigma,{{t}_{0}}+(\nu+1)\sigma]}}=0$

The latter implies the desired (4.4). Details are left to the reader. ∎

##  V Conclusion

Sufficient conditions for observability and solvability of the state
estimation for a class of nonlinear control time - varying systems are
derived. The state estimation is exhibited by means of a sequence of
functionals approximating the unknown state of the system on a given bounded
time interval. Each functional is exclusively dependent on the dynamics of
system, the input $u$ and the corresponding output $y$. The possibility of
solvability of the state estimation problem by means of hybrid observers is
briefly examined.

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Also at: ]Karazin National University, 4 Svobody sq., Kharkiv, Ukraine

# Longitudinally strong focusing lattice for Compton–ring gamma sources

Eugene Bulyak  bulyak<EMAIL_ADDRESS>[  NSC KIPT, 1
Academichna str., Kharkiv, Ukraine  Louis Rinolfi  CERN, Geneva, Switzerland
Junji Urakawa  KEK, Ibaraki, Japan

###### Abstract

Electron storage rings of GeV energy with laser pulse stacking cavities are
promising intense sources of polarized hard photons which, via pair
production, can be used to generate polarized positron beams. Dynamics of
electron bunches circulating in a storage ring and interacting with high-power
laser pulses is studied both analytically and by simulation. Common features
and differences in the behavior of bunches interacting with an extremely high
power laser pulse and with a moderate pulse are discussed. Also considerations
on particular lattice designs for Compton gamma rings are presented.

##  I Introduction. Bottle-necks.

Compton rings must be able to keep circulating intense electron bunches with a
large spread of energy of individual particles. At the same time effective
generation of the high energy photon beams demands the bunch length as short
as possible.

One of the ways to solve this problem is to employ a lattice with extremely
low momentum compaction factor (LMC) (see Araki  _et al._ (2005) and the
bibliography therein). Another known way is to make use of the longitudinal
strong focusing lattice (LSF) with a sufficiently high momentum compaction
factor and a high RF voltage (see Gallo  _et al._ (2003a) and the bibliography
therein). The idea of LSF consists in modulation of the bunch length over
ring’s circumference such that the minimal length realized at the interaction
point (IP) (naturally, the maximum is at the RF cavity).

#### To maximize yield

one has to compress the bunch both transversally and (at non head-on
collisions) longitudinally.

#### Energy spread

in Compton sources is the main draw-back:

  * •

Longitudinal one may be cured with a high rf voltage and/or a low momentum
compaction lattice;

  * •

Transversal chromaticity together with the spread leads to unstable motion of
circulating particles

#### Bunch length

is also important for the Compton rings: it should be as short as possible at
the collision point (cp) to provide the maximal yield and as long as possible
beyond cp to mitigate deteriorating interactions with the environment (walls
and joints of the vacuum chamber, etc.).

One of the way to reach proper modulation of the bunch length is employment of
the strong longitudinal focusing scheme Gallo  _et al._ (2003a) (also see
Gallo  _et al._ (2003b), Gallo  _et al._ (2004)).

The essence of the idea of double chicane scheme is following: In the low
momentum compaction scheme the ‘transparent’ insertion with high momentum
compaction (and high RF voltage as well) is introduced. We will refer to this
scheme as _the longitudinal low– $\beta$ insertion_ (LLBI) in analog to the
colliders (transversal) low–$\beta$ insertions. A scheme of LLBI is presented
in Fig. 1, a (naïve) principle of operation depicted in Fig. 2.

Figure 1: Scheme of the longitudinal low–$\beta$ insertion. RF denotes
radio–frequency cavities, M the rectangular bending magnets, IP the
interaction point(s)  Figure 2: Transformation of bunch’s longitudinal phase
portrait. 1, 5 outside of LLBI, 2 at the first RF output, 3 at IP, 4 at the
second RF input

The principle of LLBI operation is as follows. A long bunch with a small
energy spread enters the RF cavity where the particles in the head of the
bunch have been decelerated while those in the tail accelerated. Since the
pass length in the chicane formed from the rectangular magnets decreases with
energy, the bunch reaches IP with the minimal length. The second chicane and
RF cavity, symmetric to the first, restores the initial particle distribution.

In principle, the double chicane scheme represents the high–disperse ring with
a single rf cavity where the bunch length undergoes modulation such that
minimal length attained at the mid–drift, the maximal at the mid–cavity, Gallo
_et al._ (2003a). Formally LLBI reminds this scheme with the rf cavity divided
into the two and the entire ring except for chicanes and cp placed within the
rf halves. Orientation of the bunch phase portrait 1,5 in Fig.2 takes place
over the rgular part of ring’s orbit (in Gallo  _et al._ (2003a) – only in the
mid-cavity position).

##  II Model of longitudinal beam dynamics in Compton rings

In the model, a particle is being tracked on the phase space with the
coordinate axes: phase $\phi$ relative to zero voltage in the rf cavity, and
deviation of the reduced energy from the synchronous,
$p=\delta\gamma/\gamma_{s}$. Reduced time $\tau$ is counted in units of turns
of the particle over the ring: $\tau=t\beta c/L$ ($L$ is the length of the
orbit, $\beta\equiv v/c$ the reduced velocity of the synchronous particle).

Transformations of the coordinates in the lattice elements reads:

  * •

drift

$\displaystyle\phi_{f}$ $\displaystyle=$
$\displaystyle\phi_{i}+\kappa\tau_{\mathrm{dr}}p_{i}\;,$ $\displaystyle p_{f}$
$\displaystyle=$ $\displaystyle p_{i}\;;$ (1)

  * •

rf cavity

$\displaystyle\phi_{f}$ $\displaystyle=$ $\displaystyle\phi_{i}\;,$
$\displaystyle p_{f}$ $\displaystyle=$ $\displaystyle
p_{i}-U_{RF}\sin\phi_{i}\;;$ (2)

  * •

collision point

$\displaystyle\phi_{f}$ $\displaystyle=$ $\displaystyle\phi_{i}\;,$
$\displaystyle p_{f}$ $\displaystyle=$ $\displaystyle
p_{i}\left[1-b\zeta(2+p_{i})\right]-b\zeta\;.$ (3)

Here the following definitions accepted:

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle 2\pi h\eta\;;$
$\displaystyle b$ $\displaystyle=$ $\displaystyle
2(1+\cos\varphi)\gamma_{s}E_{\mathrm{las}}/E_{0}\;;$ $\displaystyle U_{RF}$
$\displaystyle=$ $\displaystyle eV/(\gamma_{s}E_{0})\;,$

with $h$ the harmonic number (ratio of the orbit length to rf wavelength);
$\tau_{\mathrm{dr}}$ ratio of the drift length to the orbit length; $V$ rf
voltage amplitude; $\eta$ the momentum compaction factor; $\zeta$ ratio of the
energy of scattered (Compton) photon to its maximal value.

##  III Linear model. Analysis.

Let us restrict our consideration to a linear case, $\sin\phi\approx\phi$,
without Compton interactions, $b=0$.

###  III.1 Small azimuthal variations.

With small azimuthal variations over turn, $|\phi_{f}-\phi_{i}|=\delta\ll 1$,
$|p_{f}-p_{i}|=\epsilon\ll 1$: low rf voltage and small momentum compaction
factor, Eqs. (• ‣ II) and (• ‣ II) are connected to Hamilton function

$H=\kappa p^{2}/2+U_{RF}\cos\phi\;,$ (4)

which represents mathematical pendulum with frequency of small oscillations
$\Omega^{2}\approx U_{RF}/\kappa$.

###  III.2 Large azimuthal variations. Single cavity.

With a high–dispersion drift and single rf cavity, strong longitudinal
focusing pattern is realized, see Gallo  _et al._ (2003b, 2004); Biscari
(2005); Falbo  _et al._ (2006).

In this case, longitudinal motion can be treated similar to transverse motion
in strong–focusing lattice of FOF/OFO type. Transformation of $(\phi,p)$
vector described by (• ‣ II), (• ‣ II) can be represented by matrices

$\displaystyle\mathbf{M}_{\mathrm{drift}}=\begin{pmatrix}1&\kappa\\\
0&1\end{pmatrix}\;,\qquad\mathbf{M}_{\mathrm{rf}}=\begin{pmatrix}1&0\\\
-U&1\end{pmatrix}\;.$ (5)

Let us write the full–turn cycle matrix as consisted of two half–drifts
$(\tau=1/2+1/2)$ and two halves of rf cavity $(U/2+U/2)$

$\displaystyle\mathbf{M}_{\mathrm{min}}$
$\displaystyle=\mathbf{M}_{\mathrm{drift/2}}\cdot\mathbf{M}_{U/2}\cdot\mathbf{M}_{U/2}\cdot\mathbf{M}_{\mathrm{drift/2}}\;;$
(6) $\displaystyle\mathbf{M}_{\mathrm{max}}$
$\displaystyle=\mathbf{M}_{U/2}\cdot\mathbf{M}_{\mathrm{drift/2}}\cdot\mathbf{M}_{\mathrm{drift/2}}\cdot\mathbf{M}_{U/2}\;.$
(7)

From any of these matrices, one can get the stability boundaries

$\left|\mathrm{Tr}\mathbf{M}\right|=\left|2-\kappa U\right|\leq 2\;,$ (8)

Thus, the stable motion can exist if

$0\leq\kappa U\leq 4\;.$ (9)

The left limit corresponds to very weak focusing (zero frequency of the
synchrotron oscillations), the right one to very strong focusing (complete
synchrotron oscillation lasts two turns).

In general, in non-uniform azimuthal focusing amplitude of synchrotron
oscillations (and the bunch length) vary along the orbit. Modulation of the
amplitude – squared ratio of minimal amplitude (in the center of the drift) to
maximal one (in the median section of rf cavity) – equal to ratio of the
matrix elements

$\mathrm{modulation}=\frac{M^{\mathrm{(min)}}_{12}}{M^{\mathrm{(max)}}_{12}}=1-\frac{\kappa
U}{4}\;.$ (10)

As expected, minimal modulation takes place for the weak focusing
($0\leftarrow\kappa U$), it is rose to full modulation at the right boundary
of stability ($\kappa U\to 4$).

###  III.3 Large azimuthal variations. Two cavities.

Making use similar procedure, we treat a lattice comprising two rf cavities
separated with two drifts. The drifts may possess different chromaticity
(phase slip factors $\kappa_{1,2}$). This scheme represents OFOFO or OFODO
pattern of focusing depending on rf cavities in-phase or in inverted phases.

Most promising case is of identical phased cavities, $U=V$, (OFOFO type).
Trace of the cyclic matrix is

$\mathrm{Tr}\mathbf{M}_{\mathrm{fof}}=2-2(\kappa_{1}+\kappa_{2})U+\kappa_{1}\kappa_{2}U^{2}\;,$
(11)

and stability region is symmetric with respect to $\kappa_{1}+\kappa_{2}$.
Designate $k=\kappa_{1}U$ and $q=\kappa_{2}U$, we get the stability region
limited with

$\displaystyle-\frac{2q}{2-q}\leq$ $\displaystyle k\leq 2\;;$
$\displaystyle-\frac{2k}{2-k}\leq$ $\displaystyle q\leq 2\;.$ (12)

The stability region is limited from top-right with straight lines (right
angle) $q=2$, $k=2$, and hyperbola $kq-2(k+q)=0$ from left-bottom. With one of
the parameters $(k,q)\to-\infty$, the interval of stability for another
becomes narrower as $2-4/q\leq k\leq 2,p\to-\infty$ (see Fig.3).

Figure 3: Stability region (red and blue curves) and lines of equal modulation
(green).

The $M_{12}$ element of the cyclic matrix at symmetric point – center of the
$\kappa_{1}$ drift (equal to $\beta\sin\mu$) – reads

$M_{12}(\tau=\kappa_{1}/2)=\frac{1}{4}(2-\kappa_{1}U)[2(\kappa_{1}+\kappa_{2})-\kappa_{1}\kappa_{2}U]\;.$
(13)

From (13) it follows the ratio of the magnitudes of enveloping function in
mid-drifts:

$\xi\equiv\frac{M_{12}(\tau=\kappa_{1}/2)}{M_{12}(\tau=\kappa_{2}/2)}=\frac{2-\kappa_{1}U}{2-\kappa_{2}U}=\frac{2-k}{2-q}\;.$
(14)

Combining (11) and (10), we can derive the stability extend for the given
modulation $\xi$:

$\displaystyle 0\leq 2-q\leq 2\left/\sqrt{\xi}\right.\;,$ $\displaystyle 0\leq
2-k\leq 2\sqrt{\xi}\;.$ (15)

#### Stability region limited by sinusoidal rf

From (14) with account (12) it may follow that to get maximal modulation of
the bunch length one must put $k\lesssim 2$ and $q\to-\infty$ if the bunch
survives in a narrow stable region. But actually, sinusoidal rf voltage (• ‣
II) puts the limits on the parameters. Really, maximal kick induced by the
cavity at $\phi=\pm\pi/2$ should cause phase change $\Delta\phi\leq\pi$.
Therefore

$\displaystyle p_{f}=-p_{i}\;;$
$\displaystyle\Delta\phi_{\mathrm{max}}=\pi\;;$
$\displaystyle|p|_{\mathrm{max}}=U/2\;;$ $\displaystyle|\kappa U|=|k|\leq
2\pi\;.$ (16)

There is a more serious consequence from this estimation: comparing the stable
interval of strong focusing (III.3) with that of weak focusing from (4), one
can see the weak focusing island of stability, $-\pi\leq\phi\leq\pi$, is two
times wider than the strong focusing one (III.3), $-\pi/2\leq\phi\leq\pi/2$.

##  IV Simulations

#### Phase trajectories

were simulated for a case of absence both synchrotron damping and Compton
interactions. With two-cavity lattice, phase trajectories were registered at
the collision point (in between chicanes and at the opposite point (middle of
the drift). In Fig.4, there presented are the phase trajectories at CP and at
mid drift, for the initial deviations of simulated particles resembling the
linear motion (small initial deviation) and the nonlinear one (large
deviation).

Figure 4: Phase trajectories in the case $2-k=0.062\;,q=-2.22$ (red color at
CP, blue at mid drift). The top graph presents small initial deviation, the
bottom one large deviations.

As it seen from the trajectories, amplitude of the envelope of phase
oscillation at CP is much smaller then at mid-drift, for the envelope of
energy deviation opposite situation holds. It worth to mention that modulation
of the phase oscillations envelope in the simulation shows fairly good
agreement with the theoretical prediction,

$\frac{\phi^{\mathrm{(max)}}_{\mathrm{cp}}}{\phi^{\mathrm{(max)}}_{\mathrm{md}}}=\sqrt{\frac{2-k}{2-q}}\approx
0.122\;.$

With increasing impact of nonlinearity, the modulation decreased (see Fig.4
the bottom panel, modulation $approx0.25$), trajectories become nonelliptic.

The nonlinear trajectories are presented in Fig.5 for the case of the same
signs of the phase slip value.

Figure 5: Phase trajectories in the case $\mathrm{sign}\;k=\mathrm{sign}\;q$
(blue color at CP, red at mid drift).

Each trajectory represents two–loop trace (for the single particle!),
orientation inverse to the case of opposite signs.

###  IV.1 Simulation of CLIC-like ring

Simulation of a Compton ring dedicated for generation of polarized gammas in
continual regime show advantages of the two–chicane scheme with opposite signs
of the phase slip factors and $q\approx-k$. Simulation revealed stability of
the scheme (no losses) and high yield of gammas as compared with the scheme
with $\mathrm{sign}\;q=\mathrm{sign}\;k$, $q\ll k$, see [PosiPol 09].

In Fig.6 and Fig.7 the mean squared bunch length and momentum spread are
plotted.

Figure 6: Mean squared bunch phase length vs. time.  Figure 7: Mean squared
energy spread vs. time.

###  IV.2 For further consideration: Compensation of $\sin\phi$ nonlinearity
by $\kappa_{3}p^{3}$ momentum compaction nonlinearity

##  V Summary and Outlook

The single-particle dynamics of electron bunches in Compton gamma sources has
been studied analytically and in simulations. The analytical estimates for the
steady-state bunch parameters are in good agreement with those obtained from
the simulations.

Our study suggests that the interaction of the electrons with the laser
photons do not significantly affect the transverse degrees of freedom. The
‘bottleneck’ is the longitudinal dynamics: the ring’s energy acceptance should
be unusually large.

## References

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  * Gallo  _et al._ (2003a) A. Gallo, P. Raimondi, and M. Zobov, in  _Workshop on e + e- in the 1-2 GeV range: Physics and Accelerator Prospects. Sept 2003, Alghero (SS) Italy_ (2003).
  * Gallo  _et al._ (2003b) A. Gallo, P. Raimondi, and M. Zobov, in  _//arXiv/physics/0309066_ (2003).
  * Gallo  _et al._ (2004) A. Gallo, P. Raimondi, and M. Zobov, in  _//arXiv/physics/0404020_ (2004).
  * Biscari (2005) C. Biscari, Phys. Rev. ST-AB 8, 091001 (2005).
  * Falbo  _et al._ (2006) L. Falbo, D. Alesini, and M. Migliorati, Phys. Rev. ST-AB 9, 094402 (2006).