Research Article | Open Access

Bogdan Sasu, "Robust Stability and Stability Radius for Variational Control Systems", *Abstract and Applied Analysis*, vol. 2008, Article ID 381791, 29 pages, 2008. https://doi.org/10.1155/2008/381791

# Robust Stability and Stability Radius for Variational Control Systems

**Academic Editor:**Stephen Clark

#### Abstract

We consider an integral variational control system on a Banach space and we study the connections between its uniform exponential stability and the stability, where and are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input-output operators acting on translation-invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.

#### 1. Introduction

In the past few years a significant progress was made in the asymptotic theory of dynamical systems and their applications in control theory (see [1–41]). It is well known that if is a Banach space, is a compact metric space, is a flow on , and is a continuous mapping, then , the solution operator of the linear differential equationis a cocycle and the pair is a linear skew-product flow. Often, (1.1) arises from the linearization of nonlinear equations (see [9, 31] and the references therein). Equation (1.1) is the starting point of our paper. We consider an integral model for the above type of systems, defined in terms of linear skew-product flows and we study the exponential stability of solutions.

Input-output conditions in the study of the asymptotic properties of evolution equations have a long and impressive history that goes back to the work of Perron. In recent years, new ideas were developed in this theory, providing input-output theorems for stability, expansiveness, and dichotomy and also a number of applications in control theory (see [2, 7, 9, 11, 13, 15, 19–27, 29, 30, 32–41] and the references therein). The aim of our paper is to establish input-output conditions for uniform exponential stability of variational control systems, identifying the viable classes for input spaces and output spaces, as well. We provide a new approach based on the fundamental properties of Banach function spaces. Our attention is devoted to the relationship between the uniform exponential stability of a variational system and the stability of a pair of spaces which are translations invariant. We establish the connections between stability and uniform exponential stability of a variational control system on a Banach space , when the input space and the output space belong to specific classes of Banach function spaces. By examples we motivate our techniques and also discuss some optimization methods with respect to the input space.

In what follows, we consider a generalization of systems described by differential equations of the formwhere is a flow on a locally compact metric space , are unbounded operators on a Banach space , and the operators , , where , are Banach spaces. Roughly speaking, the family will be subjected to additive structured perturbations, so that the perturbed system is which may be interpreted as a system obtained by applying the feedback to the time-varying system (1.2). The main question is how general may be the perturbation structure such that the property of exponential stability is preserved. We will answer this question by determining a general lower bound for the stability radius of such systems.

In the last decades, stability radius became a subject of large interest and various estimations for the lower bound of the stability radius of systems were obtained (see [11, 17–20, 32, 33, 41]). This concept was introduced by Hinrichsen, Ilchmann, and Pritchard (see [18]) and led to a systematic study of the stability of linear infinite-dimensional systems under structured time-varying perturbations. The main idea was to estimate the size of the smallest disturbance operator under which the additively perturbed system loses exponential stability. A significant theorem which gives a lower bound for the stability radius of a system associated to a mild evolution operator in terms of the norm of the input-output operator of the system has been obtained by Hinrichsen and Pritchard (see [19, Theorem 3.2]). A distinct approach of this result was given by Clark et al. in [11], employing an evolution semigroup technique. The variational case was firstly treated in [32], where we obtained a lower bound for the stability radius in terms of the Perron operators associated with a linear skew-product semiflow. In [41], Wirth and Hinrichsen introduced a concept of stability radius for the case of discrete time-varying systems under structured perturbations of multi-output feedback type. In the spirit of this theory, we have obtained in [33] an estimation for the lower bound of the stability radius of a control system of difference equations on a Banach space , under structured infinite multiperturbations, in terms of the norm of the input-output operator associated with a system on with . An interesting approach for systems in finite-dimensional spaces was presented by Jacob in [20]. There the author deduces a formula for the stability radius of time-varying systems with coefficients in , where , in terms of the norms of a family of input-output operators.

In the fourth section of the present paper we propose a new approach, deducing a lower bound for the stability radius of variational systems in terms of input-output operators acting on rearrangement invariant Banach function spaces. Until now, the most common function spaces used for estimating the stability radius are the -spaces with , which satisfy in particular the requirements of the classes introduced in this paper. Thus, our results provide a unified treatment in a large class of function spaces, extending the above-mentioned contributions. As particular cases, we obtain a lower bound for the stability radius of a variational system in terms of -spaces, with , and generalize the main result in [32].

In the last section, the central results of the paper are applied at the study of the uniform exponential stability of nonautonomous systems. We provide a complete analysis concerning the implications of the variational case for nonautonomous systems. We point out several interesting situations and deduce as particular cases the stability results due to Datko, Clark, Latushkin, Montgomery-Smith, Randolph, Neerven, Megan, and many others. Moreover, as an application we obtain a lower bound for the stability radius of nonautonomous systems in terms of the norm of the input-output operators acting on translations invariant spaces. Our results extend the existing contributions in the literature on this topic.

#### 2. Preliminary results on Banach function spaces

Let be the linear
space of all Lebesgue measurable functions , identifying the functions equal a.e.*Definition 2.1. *A linear subspace of is called *a normed function space* if there is a
mapping such that (i) if and only if a.e.;(ii), for all ;(iii), for all ;(iv) if a.e. and , then and .

If is complete,
then is called a *Banach function space.*

*Remark 2.2. *(i) If is a Banach
function space and in , then there is a subsequence such that a.e. (see, e.g.,
[28]).

(ii) If is a Banach
function space and , then .

*Definition 2.3. *A Banach function space is said to be *invariant to translations* if for every and every , if and only if
the functionbelongs to and .

Let be the linear space of all continuous functions with compact support contained in and let . Let be the linear space of all locally integrable functions .

We denote by the class of all Banach function spaces which are invariant to translations and with the following properties:(i);(ii)if , then there is a continuous function

For every we denote by the characteristic function of the set .

*Remark 2.4. * If , then , for all .

Indeed, let . If is a continuous function with compact support such that , for and , for , then we have that , for all . Since , using (iv) from Definition 2.1, we deduce that .

*Definition 2.5. *If , then the function , is called *the fundamental function* of the space .

*Remark 2.6. * The function is
nondecreasing.

*Example 2.7. *Let . Then the space with respect to
the norm is a Banach
function space which belongs to .*Example 2.8. *Let be the linear
space of all essentially bounded functions . With respect to the norm , is a Banach
function space which belongs to .*Example 2.9 (Orlicz spaces). * Let be a
nondecreasing left-continuous function, which is not identically zero on . The *Young
function* associated with is defined by For every we define The set of all with the
property that there is such that is a linear
space. With respect to the norm , is a Banach
space called *the Orlicz space* associated with .

*Remark 2.10. * It is easy to verify that the Orlicz spaces
introduced in Example 2.9 belong to .

*Remark 2.11. * The spaces , are Orlicz
spaces, which may be obtained for , if and for

Lemma 2.12. *Let and . Then the function , , belongs to .*

*Proof. * We have thatThis shows that and

*Definition 2.13. *Let . Say that and are *equimeasurable* if for every the sets and have the same
measure.*Definition 2.14. *A Banach function space is *rearrangement invariant* if for every
equimeasurable functions with , one has and .

*Remark 2.15. * The Orlicz spaces are rearrangement invariant (see
[4, Theorem 8.9]).

We denote by the class of all Banach function spaces which are rearrangement invariant.

*Remark 2.16. * If , then is an
interpolation space between and (see [4, Theorem 2.2, page
106]).

Lemma 2.17. *Let and . Then for every the function**belongs to . Moreover, there is such that , for all .*

*Proof. * It is easy to see thatis a correctly defined bounded
linear operator. Moreover, the restriction is correctly
defined and is a bounded linear operator. Then from Remark 2.16 it follows that is a correctly
defined and bounded linear operator. Taking , the proof is complete.

We denote by the class of all Banach function spaces with the property that

*Remark 2.18. * If , for all , then the Orlicz space (see, e.g.,
[24, Proposition
2.1]).

Let be the space of
all continuous functions with .
Lemma 2.19. *If , then .*

*Proof. * Let . Then there is an unbounded increasing sequence such that , for all and . We set . According to our hypothesis, . Then, we observe thatHence, the sequence is fundamental
in , so there is such that in . From Remark 2.2 there exists a subsequence such that a.e. This
implies that a.e., so in . It follows that and the proof
is complete.

We denote by the class of all Banach function spaces with the property that .

*Remark 2.20. *Using Remark 2.2(ii) we have that if , then there is a continuous function with .

Lemma 2.21. *If , then or .*

*Proof. * Suppose by contrary that there exists such that and . Then and . It follows that there is such that , for all . In particular, for we deduce that , for all , which is absurd.

Let be a real or complex Banach space. We denote by the linear space of all continuous functions with compact support, let and let be the linear space of all locally integrable functions .

For every we denote by the linear space of all Bochner measurable functions with the property that the mapping , lies in . Endowed with the norm , is a Banach space.

#### 3. Stability of variational integral control systems

In the stability theory of evolution operators in Banach spaces, an evolution family is said to be stable if for every the mapping belongs to whereThe classical input-output results concerning exponential stability of evolution families in Banach spaces may be stated as follows.

*Let**be an evolution
operator on a Banach space X and**with*. * Then the following assertions are equivalent:*(i)*is uniformly exponentially stable;*(ii)*is**stable;*(iii)*is**stable;*(iv)*is**stable.*

The implication (iv)(i) was obtained by Datko in 1973 (see [16]), using one of his stability results contained in the same paper. In the last few years, there were pointed out new and interesting methods of proving the above theorem. The equivalences (i)(ii)(iv) were proved by Neerven for the special case of semigroups of linear operators (see [30]) and . In [11], Clark et al. have provided an inedit technique of proving the equivalences (i)(ii)(iv) for the case . There the authors established important connections between the asymptotic properties of evolution families and those of the associated evolution semigroups with a large number of applications in control theory. Roughly speaking the authors described the behavior of a nonautonomous system in terms of the properties of an associated autonomous system. Therefore, in their approach the input space must coincide with the output space.

The equivalence (i)(ii)(iii) has been also proved by van Minh et al. (see [29]), employing an evolution semigroup technique. The equivalence (i)(ii) was also treated by Buşe in [7]. The case was completely treated in [23], where several new stability results of Perron type were established for exponential stability of evolution families, generalizing the above equivalences and obtaining that uniform exponential stability of an evolution family can be expressed using boundedly locally dense subsets of and , respectively. The main results in [23] were extended in [25] for the case of linear skew-product flows, where the input-output operator was replaced with a family of operators acting between the function spaces of the admissible pair. The stability results obtained in [25] generalized some theorems from [7, 16, 23, 29, 30]. Moreover, the stability theorems in [25] led to several interesting consequences in control theory (see [26, 32]).

In what follows, the input-output techniques in the stability theory of variational equations will be treated from the perspective of Banach function spaces arising from the interpolation theory. In this section, our main purpose is to give a complete and unified study of the exponential stability of variational systems via input-output methods, providing the “structure” of the classes of input and output spaces, respectively.

Let be a Banach
space, let be a metric
space, and let . We denote by the Banach
algebra of all bounded linear operators on . The norm on and on will be denoted
by .
*Definition 3.1. *Let . A continuous mapping is called *a flow* on if and , for all .*Definition 3.2. *A pair is called a *linear skew-product flow* on if is a flow on and satisfies the
following conditions:(i), the identity operator on , for all ;(ii), for all (*the
cocycle identity*);(iii) is continuous,
for every (iv) there are and such that , for all

The mapping given by
Definition 3.2 is called *the cocycle* associated to the linear skew-product flow .

Let be a linear
skew-product flow on . In what follows, we consider the variational
integral control systemwith and .
*Definition 3.3. *The system () is said to be(i)*uniformly stable* if there is such that(ii)*uniformly exponentially stable* if there
are such that

*Remark 3.4. * (i) The system () is uniformly
stable if and only if there is such that , for all .

(ii) The system () is uniformly exponentially stable if and only if there are such that , for all .

*Definition 3.5. *Let , be two Banach
function spaces with . The system () is said to be *stable* if the
following assertions hold:(i)for every and every the solution (ii)there is such that , for all . is called * the input space* and is called *the output space*.

We begin with a sufficient condition for uniform
stability.
Theorem 3.6. *Let . If the system () is stable, then
the system () is uniformly
stable.*

*Proof. * Let be a continuous
function with and .

Let be given by
Definition 3.2 and let be given by
Definition 3.5.

Let . We consider the function , We have that and , for all . This implies thatFrom hypothesis we have that andWe observe that Then, for we deduce thatwhich implies thatSince is invariant to
translations we have that . Then, from relations (3.4)–(3.8) we obtain thatFor we have that . Then, for we deduce that , for all . Taking into account that does not depend
on or , it follows that , for all and all and the proof
is complete.

The first main result of this section is the following.

Theorem 3.7. *Let . If and the system () is stable, then
the system () is uniformly
exponentially stable.*

*Proof. * Let be given by
Definition 3.5. From Theorem 3.6 we have that there is such that for all

Since from Remark 2.20
it follows that there is a continuous function such that . Let be such that

For every let be a continuous
function with , for , , and , for . Sincethere is such that

Let . We consider the functionWe have that , so . In addition, for all . This implies that We observe that Then, we have thatwhich implies thatSince is invariant to
translations we have that . Then from (3.16) we deduce thatAccording to our hypothesisUsing relations (3.17) and (3.18)
it follows thatWe set . From (3.19) and (3.12) we obtain that . Since does not depend
on or we deduce that , for all .

Let and . Let . Then there is and such that . It follows that . This implies that the system () is uniformly
exponentially stable.

The second main result of this section is the following.

Theorem 3.8. *Let . If and the system () is stable, then () is uniformly
exponentially stable.*

*Proof. * Let be given by
Definition 3.5 and let be given by
Theorem 3.6. Let be such thatLet be a continuous
function with and

Let . We consider the functionSince we have that . Moreover, , for all , which implies thatWe observe that , for all . Then, we have that Since we have that . Then from (3.23) and (3.22) we deduce that From relations (3.20) and (3.24)
it follows that . Taking into account that does not depend
on or we obtain that , for all . Using similar arguments as in Theorem 3.7 we obtain
that the system () is uniformly
exponentially stable.

The central result of this section is the following.

Theorem 3.9. *Let be such that or . Then, the following assertions hold:*(i)*if the
system () is stable, then () is uniformly
exponentially stable;*(ii)*if and one of the
spaces , belongs to the
class , then the system () is uniformly
exponentially stable if and only if the system () is stable.*

*Proof. * (i) This follows from Theorems 3.7 and 3.8.

(ii) *Necessity.* Let be such that , for all and all . Since , there is such that

Let and letThen for every we have that*Case 1. * Let be given by
Lemma 2.17. Since , from Lemma 2.17 we have that and . Then from (3.27) it follows that , so . Moreover, from (3.25) and (3.27) we have thatTaking and observing
that does not depend
on or we deduce that
the system () is stable.*Case 2. * Let be given by
Lemma 2.17. From , using Lemma 2.17 we have that and . Then, from (3.25) and (3.27) we deduce that andTaking we conclude
that the system () is stable.*Sufficiency.* It follows from (i).

In what follows, we prove that the main result given
by Theorem 3.9 is the most general in this topic. Specifically, we will show
that if and , then the stability of
the system () does not imply
the uniform exponential stability of ().

*Example 3.10. *Let and let , . Let and for every , letThen is a linear
skew-product flow on . We consider the systemwith and . Observing that , for all , we deduce that the system () is not
uniformly exponentially stable.

Let with and . Then and from Lemma
2.19 we have that . In what follows, we prove that the system () is stable.

Let and . Let . Since , there is such that . Let From we have that
there is such that , for all . Then, for , we deduce thatThis implies that so . It follows that for every we have that .

It makes sense to define the linear operatorIt is easy to verify that is closed, so
from the closed graph theorem we obtain that is bounded.
Setting we have thatSince there is such thatSince there is such that

Let . Since and it follows that . It is easy to observe that . Setting , from relations (3.34)–(3.36) we deduce that Taking into account that does not depend
on or we conclude
that the system () is stable. But,
for all that () is not
uniformly exponentially stable.

*Remark 3.11. *
The input-output characterizations for asymptotic
properties of systems have a wide applicability area if the input space is as
small as possible and the output space is very general. We note that in the
main result given by Theorem 3.9 the input functions belong to , while the output space is a general function space.

Moreover, the class is closed to
finite intersections. If , then we may consider the space with respect to
the normwhich is a Banach function space
in . Now, if we analyze condition (ii) in Definition 3.5,
it is obvious that if the input space is , then in the right member of the inequality we have a
“larger” norm and thus the estimation is more permissive.

Let , , and . We consider the spacewhich is a Banach space with
respect the normCorollary 3.12. * The
system () is uniformly
exponentially stable if and only if the system () is stable.*

*Proof. * If , then . If , then . By applying Theorem 3.9 we obtain the
conclusion.

Corollary 3.13. *Let . The system () is uniformly
exponentially stable if and only if () is stable.*

*Proof. * This follows from Theorem 3.9 and Lemma 2.21.

Lemma 3.14. *Let and with . For every , the function**belongs to . In addition, there is such that , for all .*

*Proof. * Using Hölder's inequality it follows that for every the function . Then, fromwe deduce that for every the function belongs to . Hence, it makes sense to define the linear operator , . It is easy to see that is closed, so
it is bounded. Setting we obtain the
conclusion.

Corollary 3.15. *Let with . The following assertions hold:*(i)*if the system () is stable, then () is uniformly
exponentially stable;*(ii)* if , then the system () is uniformly
exponentially stable if and only if it is stable.*

*Proof. * (i) It follows from Theorem 3.9(i).

(ii) *Necessity.* Let be given by
Lemma 3.14. Let be such that , for all and all . Let . ThenUsing Lemma 3.14 we obtain that andTaking into account that and do not depend
on or , it follows that the system () is stable.*Sufficiency.* This follows from (i).

*Definition 3.16. *Let , be two Banach
function spaces with . The system () is said to be *completely**stable* if the following assertions hold:(i)for every and every the solution (ii)there is such that , for all .Theorem 3.17. *Let be such that or . Then, the following assertions hold:*(i)* if the system () is completely stable, then () is uniformly
exponentially stable;*(ii)* if and one of the
spaces , belongs to the
class , then the system () is uniformly
exponentially stable if and only if the system () is completely stable.*

*Proof. * (i) This follows from Theorem 3.9(i).

(ii) *Necessity* follows using similar arguments as in the necessity part of Theorem 3.9(ii).*Sufficiency* is given by (i).

Corollary 3.18. *Let , and . The system () is uniformly
exponentially stable if and only if the system () is completely stable.*

*Proof. * This follows from Corollary 3.12.

Corollary 3.19. *Let . The system () is uniformly
exponentially stable if and only if () is completely stable.*

*Remark 3.20. * Let . If the system () is uniformly
exponentially stable, then for every the linear
operatoris correctly defined and
bounded. Moreover, if is given by
Definition 3.16, then we have that .

Corollary 3.21. *Let with . The following assertions hold:*(i)* if the system () is completely stable, then () is uniformly
expo*