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Abstract and Applied Analysis
Volume 2009 (2009), Article ID 670314, 34 pages
http://dx.doi.org/10.1155/2009/670314
Research Article

On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays

IIDP, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), Aptdo, 644-Bilbao, Spain

Received 5 March 2009; Accepted 26 May 2009

Academic Editor: Ülle Kotta

Copyright © 2009 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressions are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.

1. Introduction

The stabilization of dynamic systems is a very important issue since it is the first requirement for most of the applications. Powerful techniques for studying the stability of dynamic systems are Lyapunov stability theory and fixed point theory which can be easily extended from the linear time invariant case to the time varying one as well as to functional differential equations, as those arising for instance from the presence of internal delays, and to certain classes of nonlinear systems [1, 2]. Dynamic systems which are of increasing interest are the so-called switched systems which consist of a set of individual parameterizations and a switching function which selects along time the particular parameterization to be activated during a subsequent time interval. Switched systems are essentially time varying by nature even if all the individual parameterizations are time invariant due to the operation mode of the switching function. The major interest of such systems arises from the fact that some real word existing systems are able to change their parameterizations to better adapt to their environments. Another important interest of some of such systems relies on the fact that changes of parameterizations through time can lead to benefits in certain applications while maintaining global stability [313]. The properties of uniform exponential stability, robust exponential stability, and ultimate boundedness are very important in dynamic systems as discussed in [4] under structured perturbations in the context of a variational control system. The interest of stabilization of dynamic systems has been of interest in many applications including, for instance, optimal control, switching control, switched dynamic systems, systems parameterized within polytopes, and functional systems including time delay systems, see, for instance, [627]. On the other hand, time delay dynamic systems are very important in the real life for appropriate modelling of certain biological and ecological systems, and they are present in physical processes implying diffusion, transmission, teleoperation, population dynamics, war and peace models, and so forth (see, for instance, [1, 2, 1218]). Linear switched dynamic systems are a very particular case of the dynamic system proposed in this manuscript. A switched system can result, for instance, from the use of a multimodel scheme, a multicontroller scheme, a buffer system, or a multiestimation scheme (see, for instance, [3, 17, 1924, 28, 29]). For instance, a (nonexhaustive) list of papers with deal with some of these questions related to switched systems follows.

(1)In [15], the problem of delay-dependent stabilization for singular systems with multiple internal and external incommensurate delays is focused on. Multiple memory-less state feedback controls are designed so that the resulting closed-loop system is regular independent of delays, impulse free, and asymptotically stable. (2)In [28], the problem of the 𝑁-buffer switched flow networks is discussed based on a theorem on positive topological entropy. (3)In [19], a multimodel scheme is used for the regulation of the transient regime occurring between stable operation points of a tunnel diode-based triggering circuit. (4)In [20, 21], a parallel multiestimation scheme is derived to achieve close-loop stabilization in robotic manipulators whose parameters are not perfectly known. The multiestimation scheme allows the improvement of the transient regime compared to the use of a single estimation scheme while achieving at the same time closed-loop stability. (5)In [22], a parallel multiestimation scheme allows the achievement of an order reduction of the system prior to the controller synthesis so that this one is of reduced order (then less complex) while maintaining closed-loop stability. (6)In [23], the stabilization of switched dynamic systems is discussed through topologic considerations via graph theory. (7)The stability of different kinds of switched systems subject to delays has been investigated in [1113, 17, 24, 29]. (8)The stability switch and Hopf bifurcation for a diffusive prey-predator system is discussed in [6] in the presence of delay. (9)A general theory with discussed examples concerning dynamic switched systems is provided in [3].

A class of integrodifferential impulsive periodic systems is investigated in [5] on a Banach space through an impulsive periodic evolution operator. The results in this paper emphasize the importance of evolution operators for analysis of the solution of integrodifferential systems. The dynamic system under investigation is a linear switched system subject to internal point delays and feedback state-dependent impulsive controls which is based on a finite set of time varying parametrical configurations and switching function which decides which parameterization is active during a time interval as well as the next switching time instant. Explicit expressions for the state and output trajectories are provided together with the evolution operators and the input-state and input-output operators under zero initial conditions. The causal and anticausal Toeplitz as well as the causal and anticausal Hankel operators are defined explicitly for the case when all the configurations have auxiliary unforced delay-free systems being dichotomic (i.e., with no eigenvalues on the complex imaginary axis); the controls are square-integrable, and the input-output operators are bounded. It is proven that if the anticausal Hankel operator is zero independent of the delays and the system is uniformly controllable and uniformly observable independent of the delays then the system is globally asymptotically Lyapunov's stable independent of the delays. Those results generalize considerably some previous parallel background ones for the delay-free and switching-free linear time- invariant case [25]. The paper is organized as follows. Section 2 discusses the various evolution operators valid to build the state-trajectory solutions in the presence of internal delays and switching functions operating over a set of time invariant prefixed configurations. Stability and instability are discussed from Gronwall's lemma [29] for the case when the auxiliary unforced delay-free system possesses only dichotomic time invariant configurations. Analytic expressions are given to define such operators as well as the input-state and input-output ones under zero initial conditions. Section 3 discusses the input-state and input-output and operators if the input is square-integrable and the state and output are also square-integrable. Related to those operators proved to be bounded under certain condition, the causal and anticausal state-input and state-output Hankel and the causal and anticausal state-input and state-output Toeplitz operators are defined explicitly. The boundedness of the state-input/output operators is proven if the controls are square-integrable and the matrices of all the active configurations of the auxiliary-delay free system are dichotomic for the given switching function. The causality and anticausality of the switched system are characterized, and some relationships between the properties of causality, stability, controllability, and observability are also proven.

Notation 1. 𝐙,𝐑, and 𝐂 are the sets of integer, real, and complex numbers, respectively.
𝐙+and𝐑+ denote the positive subsets of 𝐙, respectively, and 𝐂+ denotes the subset of 𝐂 of complex numbers with positive real part.
𝐙and𝐑 denote the negative subsets of 𝐙, respectively, and 𝐂 denotes the subset of 𝐂 of complex numbers with negative real part.

Given some linear space 𝑋 (usually 𝐑 or 𝐂) then 𝐶(𝑖)(𝐑0+,𝑋) denotes the set of functions of class 𝐶(𝑖). Also, BPC(𝑖)(𝐑0+,𝑋) and PC(𝑖)(𝐑0+,𝑋) denote the set of functions in 𝐶(𝑖1)(𝐑0+,𝑋) which, furthermore, possess bounded piecewise continuous constant or, respectively, piecewise continuous constant 𝑖th derivative on 𝑋.

The set of linear operators from the linear space 𝑋 to the linear space 𝑌 are denoted by 𝐋(𝑋,𝑌), and the Hilbert space of 𝑛 norm-square Lebesgue integrable real functions on 𝐑 is denoted by 𝐋𝑛2𝐋𝑛2(𝐑) and endowed with the inner product 𝐿2-norm 𝑓𝐋𝑛2=(𝑓(𝜏)22𝑑𝜏), for all 𝑓𝐋𝑛2, where 2 is the 2-vector (or Euclidean) norm and its corresponding induced matrix norm. 𝐋𝑛2[𝛼,) the Hilbert space of 𝑛 norm-square Lebesgue integrable real functions on [𝛼,)𝐑 for a given 𝛼𝐑 which is endowed with the norm 𝑓𝐋𝑛2[𝛼,)=(𝛼𝑓(𝜏)22𝑑𝜏), for all 𝑓𝐋𝑛2[𝛼,). 𝐋𝑛2+={𝑓𝐋𝑛2𝑓(𝑡)=0,forall𝑡𝐑} and 𝐋𝑛2={𝑓𝐋𝑛2𝑓(𝑡)=0,forall𝑡𝐑+} are closed subspaces of 𝐋𝑛𝟐={𝑓𝐋𝑛2𝑓(𝑡)=0,forall𝑡𝐑}𝐋𝑛𝟐 of respective supports 𝐑0+ and 𝐑0. Then, 𝐋𝑛2=𝐋𝑛2+𝐋𝑛2.

𝐼𝑛 denotes the 𝑛th identity matrix.

𝜆max(𝑀) and 𝜆min(𝑀) stand for the maximum and minimum eigenvalues of a definite square real matrix 𝑀=(𝑚𝑖𝑗).

𝜎𝐑0+𝑁={1,2,,𝑁} is the switching function which defines the parameterization at time 𝑡 of a switched dynamic system among 𝑁 possible time invariant parameterizations. 𝜎𝜏,𝑡(=𝜎[0,t))[0,t)(𝐑0+)𝑁𝜏,𝑡𝑁 is the partial switching function with its domain restricted to [𝜏,𝑡]. 𝜎𝑡is a notational abbreviation of 𝜎0,𝑡.

The point constant delays are denoted by 𝑖[0,],forall𝑖𝑞{0} and are, in general, incommensurate, and 0=0.

2. The Dynamic System Subject to Time Delays

Consider the following class of switched linear time-varying differential dynamic system subject to 𝑞 distinct internal incommensurate point delays 0=0<1<2<<𝑞=:

where 𝑖(0,); for all 𝑖𝑞={1,2,,𝑞}, 𝑥(𝑡)𝐑𝑛, 𝑢(𝑡)𝐑𝑚, and 𝑦(𝑡)𝐑𝑝are the state, input (or control) and output (or measurement) vectors, respectively, and

where 𝑖𝑞{0}={0,1,2,,𝑁}, fulfilling that 𝐴𝑖(𝜏), 𝐵𝑖(𝜏), 𝐶𝑖(𝜏) and 𝐷𝑖(𝜏) are piecewise constant such that they are constant either in (𝑡𝑇,𝑡] or in [𝑡,𝑡+𝑇), for all 𝑡𝐑0+ and some fixed 𝑇𝐑+. The system (2.1) has two auxiliary unforced systems which are useful for stability analysis defined as follows.

(i)The zero-delay auxiliary unforced switched system (2.1):̇𝑥(𝑡)=(𝑞𝑖=0𝐴𝑖(𝑡))𝑥(𝑡); 𝑦(𝑡)=𝐶(𝑡)𝑥(𝑡) is the particular system arising when all the delays of (2.1) are zero. (ii)The delay-free unforced auxiliary switched system:  ̇𝑥(𝑡)=𝐴0(𝑡)𝑥(𝑡𝑖);  𝑦(𝑡)=𝐶(𝑡)𝑥(𝑡) is the particular system arising when all the matrices describing delayed dynamics in (2.1) are zero.

A well known important property is that, in the case of one single configuration, (i.e., the system does not switch among a set of them) the global stability of the above auxiliary systems leads to necessary conditions for stability independent of the delays [26]. The physical interpretation is that the dynamic system (2.1) is a switched system under some (piecewise constant) switching function 𝜎𝐑0+𝑁, which generates a strictly ordered sequence of switching time instants 𝑆𝑇𝜎={𝑡𝑖𝑡𝑖+1𝑡𝑖+𝑇,forall𝑖𝑁0({1})𝐙+,𝑡1𝐑0+}, and which might be equivalently rewritten, since 𝐴𝑖(𝑡)=𝐴𝑖𝜎(𝑡), for all 𝑖𝑁{0}, 𝐵(𝑡)=𝐵𝜎(𝑡), 𝐶(𝑡)=𝐶𝜎(𝑡), 𝐷(𝑡)=𝐷𝜎(𝑡) via the switching function 𝜎𝐑0+𝑁, as

where 𝐑0+ is the state-trajectory solution, which is almost everywhere time differentiable on [,0) and satisfies (2.3), subject to bounded piecewise continuous initial conditions on 𝑥=𝜑BPC(0)([,0],𝐑𝑛), that is, 𝜎(𝑡)=𝑗𝑁,forall𝑡𝐑[0,𝑡1). It is assumed that 𝑡1ST𝜎, 𝜎𝐑0+𝑁, being the first switching instant generated by the switching function (,𝑡1]; that is, there is a time invariant parameterization belonging to the given set on 𝐑. The above assumption has an obvious real meaning for the general cases where the control is nonzero on𝐑0+. The unique mild solution of the state-trajectory solution, which exists on 𝜑BPC(0)([,0],𝐑𝑛) according to Picard-Lindeloff theorem for any given 𝑢BPC(0)(𝐑,𝐑𝑚) and any [𝛼,𝑡]𝐑, may be calculated on any time interval []=𝛼,𝑡𝛼,𝑡𝑘𝑖𝑁𝑡(𝛼)𝑡𝑘+𝑖,𝑡𝑘+𝑖+1𝑡𝑘+1+𝑁𝑡(𝛼),𝑡,(2.5) on nonzero measure by first decomposing the interval as a disjoint union of connected components defined by its contained sequence of switching time instants as

where 𝑁𝑡(𝛼)={𝑖𝐍SI𝜎𝑡𝑖𝑡}; 𝑡𝑘+𝑖ST𝜎, 𝑖𝑁𝑡(𝛼), for all 𝑡𝑘+𝑁t(𝜎)+1ST𝜎 and 𝜎(𝑡𝑘)=𝑗(𝑡𝑘)𝜎(𝑡+𝑘)=𝑗(𝑡𝑘+1)𝑁. Note that 𝑡𝑘,𝑡𝑘+1ST𝜎, for all 𝑥(𝑡)=𝚽𝑥(𝛼)(𝑡)+𝚪𝑢𝛼(𝑡)=Φ(𝑡,𝛼)𝑥(𝛼)+𝑞𝑖=1𝑡𝛼Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑥𝜏𝑖𝑑𝜏+𝑡𝛼Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏,(2.6). Then, the state trajectory solution is

where, although the evolution operators between any two time instants 𝜎𝜏,𝑡 depends on the corresponding partial switching function Φ(𝑡,𝜏), the simpler notation Φ𝜎𝜏,𝑡(𝑡,𝜏) is preferred instead for 𝑀𝜎(𝑡)𝑀(𝑡) for the sake of simplicity. This simplified notation criterion will be used when no confusion is expected together with the former one 𝑦(𝑡)=𝐶𝚽𝑥(𝛼)(𝑡)+𝐶𝚪𝑢𝛼+𝐷(𝑡)=𝐶(𝑡)Φ(t,𝛼)𝑥(𝛼)+𝑞𝑖=1𝑡𝛼Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑥𝜏𝑖𝑑𝜏+𝑡𝛼Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+D(𝑡)u(𝑡),(2.7) for all the matrices of the individual parameterizations. The output trajectory solution is

for all 𝜑BPC(0)([,0],𝐑𝑛), subject to initial conditions 𝑥(𝛼), where

(1) [𝛼,] is the strip of state-trajectory solution on 𝜑(𝑡) which takes values 𝑡=𝛼<0 if Φ𝐋(𝐑𝑛×𝐑,𝐑𝑛)

(2) the evolution operator in 𝚽𝑥(𝛼)(𝑡)=Φ(𝑡,𝛼)𝑥(𝛼)+𝑞𝑖=1𝑡𝛼Φ(𝑡,𝜏)𝑥𝜏𝑖𝑑𝜏,𝑡(𝛼),𝛼𝐑,(2.8) is defined pointwisely by

so that [0,𝑡] is the unforced response in Φ𝐶(0)(𝐑×𝐑,𝐑𝑛×𝑛), where the matrix function 𝐑 is a fundamental matrix of the dynamic differential system which is everywhere differentiable and has almost everywhere continuous time-derivative on ST𝜎with bounded discontinuities on the set [𝛼,𝑡]𝐑 and is defined on the interval Φ(𝑡,𝛼)=𝑒𝐴0(𝑡𝑘+𝑁𝑡(𝛼)+1)(𝑡𝑡𝑘+𝑁𝑡(𝛼)+1)𝑘+𝑁𝑡(𝛼)𝑖=1𝑒𝐴0𝑡𝑘+𝑖𝑡𝑘+𝑖+1𝑡𝑘+𝑖𝑒𝐴0𝑡(𝛼)𝑘𝛼,(2.9) as

and the above matrix function products are defined to the left, and

(3) the input-state and input-output operators in Γ𝑜𝐋(𝐑𝑚×𝐑,𝐑𝑝)and Γ𝑜=𝐶𝜎Γ+𝐷𝜎, respectively, 𝚪𝑢𝛼𝑡(𝑡)=𝑡𝛼Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏=𝑡Φ(𝑡,𝜏)𝐵(𝜏)𝑢𝛼𝑡(𝚪𝜏)𝑑𝜏𝑜𝑢𝛼𝑡(𝑡)=𝑡𝛼=𝐶(𝑡)Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)𝑡𝐶(𝑡)Φ(𝑡,𝜏)𝐵(𝜏)𝑢𝛼𝑡(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),𝑡(𝛼),𝛼𝐑0+,(2.10), are defined pointwisely by

where

so that

are, respectively, the unforced state and output responses in 𝑥(𝑡)=𝑍(𝑡,𝛼)𝑥(𝛼)+𝑞𝑖=1𝛼𝛼𝑖𝑍(𝑡,𝜏)𝑥(𝜏)𝑑𝜏+𝑡𝑍(𝑡,𝜏)𝐵(𝜏)𝑢𝛼𝑡(𝜏)𝑑𝜏,𝑦(𝑡)=𝐶(𝑡)𝑍(𝑡,𝛼)𝑥(𝛼)+𝑞𝑖=1𝛼𝛼𝑖𝑍(𝑡,𝜏)𝑥(𝜏)𝑑𝜏+𝑡𝑍(𝑡,𝜏)𝐵(𝜏)𝑢𝛼𝑡(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),(2.13). The state and output trajectory solutions (2.6), or (2.7), under (2.8)–(2.10), subject to the output equation in (2.1) are identically defined by with initial conditions 𝑥(0)=𝜑(0) so that 𝑍(𝑡,𝛼)𝐶(0)(𝐑×𝐑,𝐑𝑛×𝑛), 𝐑+ is an everywhere differentiable matrix function on SI𝜎, with almost everywhere continuous time-derivative except at time instants in ̇𝑍(𝑡)=𝑞𝑖=0𝐴(𝑡)𝑍𝑡𝑖,0(2.14), which satisfies

on 𝑍(𝑡,𝛼)=0 whose unique solution satisfies 𝛼(<𝑡), for all 𝑡𝐑, 𝐼𝑍(𝑡,𝛼)=Φ(𝑡,𝛼)𝑛+𝑞𝑖=1𝑡𝛼Φ(𝛼,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖,𝛼𝑑𝜏,𝑡(𝛼),𝛼𝐑(2.15), and is defined by

on any time interval 𝛼=0. Now, take 𝑢(𝑡), and consider that the input 𝐑 is defined on 𝐼𝑥(𝑡)=Φ(t,0)𝑛+𝑞𝑖=1𝑡0Φ(0,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖+,0𝑑𝜏𝑥(0)𝑞𝑞𝑖=1𝑗=1t00𝑗Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖+,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏𝑡Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝑞𝑖=1𝑡0𝜏𝑖Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖𝐼,𝛾𝐵(𝛾)𝑢(𝛾)𝑑𝛾𝑑𝜏=Φ(𝑡,0)𝑛+𝑞𝑖=1𝑡0Φ(0,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖+,0𝑑𝜏𝑥(0)𝑞𝑞𝑖=1𝑗=1𝑡00𝑗Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖𝜑+,𝛾(𝛾)𝑑𝛾𝑑𝜏𝑡Φ(𝑡,𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖,𝜏𝐵(𝜏)𝑈(𝜏)𝑈𝛾𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏,𝑦(𝑡)=𝐶𝜎(𝑡)𝐼Φ(𝑡,0)𝑛+𝑞𝑖=1𝑡0Φ(0,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖+,0𝑑𝜏𝑥(0)𝑞𝑞𝑖=1𝑗=1𝑡00𝑗Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏+𝑡+Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑞𝑖=1𝑡0𝜏𝑖Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖Φ𝐼,𝛾𝐵(𝛾)𝑢(𝛾)𝑑𝛾𝑑𝜏+𝐷(𝑡)𝑢(𝑡)=𝐶(𝑡)(𝑡,0)𝑛+𝑞𝑖=1𝑡0Φ(0,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖𝑥+,0𝑑𝜏(0)𝑞𝑞𝑖=1𝑗=1𝑡00𝑗Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖+,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏𝑡Φ(𝑡,𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖,𝜏×𝐵(𝜏)𝑈(𝜏)𝑈𝛾𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),(2.16). Then, the combination of (2.7) with the substitution of (2.13) in the delayed state and output-trajectory solutions yields

where 𝜎𝐑0+N is the unit step (Heaviside) function. The following result is concerned with sufficient conditions of asymptotic stability and exponential stability of the switched delayed system (2.1), (2.3), based on Gronwall's lemma, which will be then useful to define the Hankel and Toeplitz operators.

Theorem 2.1. The following properties hold.
(i) The unforced dynamic system (2.1), (2.3) is globally asymptotically stable independent of the sizes of the delays if the switching function lim𝑡𝑡𝑖𝑆𝑇𝜎(𝑡)𝐾0𝑥0200𝑑𝑡0𝜎𝑖𝑒1+𝜌0𝜎𝑡𝑖1𝜌𝑡0𝜎𝑖𝑞𝑖=1𝐴𝑡𝑖𝜎𝑖2×𝑒(𝜌𝑖)0𝜎(𝑡𝐾𝑖)0𝜎(𝑡𝑞𝑖=1||𝐴𝑖)𝑖𝜎(𝑡||2)𝑡𝑖+1𝑡𝑖=0,(2.17) is such that where 𝐑+϶𝜌0𝜎(𝑡𝑖){𝜌01,𝜌02,,𝜌0𝑁} and 𝜎(𝑡𝑖)=𝑗𝑁 if 𝑒𝐴0𝑖𝑡2𝐾0𝑖𝑒𝜌0𝑖𝑡,forall𝑖𝑁 are real constants such that 𝐀0 (i.e., all the matrices in the set 𝑆𝑇𝜎(𝑡)={𝑡𝑖𝑆𝑇𝜎𝑡𝑖𝑡} are stable) with 𝑡𝑠(𝑡)+1=𝑡 and 𝑡𝑆𝑇𝜎 if s(𝑡)=card𝑆𝑇𝜎(𝑡), where 𝜎𝐑0+𝑁.
(ii) The unforced dynamic system (2.1), (2.3) is globally exponentially stable independent of the sizes of the delays if the switching function 𝐴0𝑗 is such that 𝜌0𝑗>𝐾0𝜎(𝑡𝑖)𝑞𝑖=1𝐴𝑖𝑗2are all stable matrices satisfying 𝑗𝑁, for all max𝑡𝑖𝑆𝑇𝜎(𝑡𝑖+1𝑡𝑖)𝑇, and the residence time at each switching instant satisfies |𝜌0𝑗| with its lower-bound T being sufficiently large according to the respective absolute values 𝐴0𝑗 of the stability (or convergence) abscissas of 𝜌0𝑗<0 (i.e., 𝐴0𝑗 if all the eigenvalues of 𝜌0𝑗+𝜀,𝜀0+ are distinct and 𝑗𝑁, otherwise), for all 𝐴𝑖𝑗(forall𝑖𝑞,𝑗𝑁) and the norms of the matrices 𝜎𝐑0+𝑁.
(iii) The unforced dynamic system (2.1), (2.3) is globally exponentially stable independent of the sizes of the delays if the switching function 𝐴0𝑗 is such that at least one 𝜌0𝑗>𝐾0𝜎(𝑡𝑖)𝑞𝑖=1𝐴𝑖𝑗2is a stable matrix satisfying max𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎(𝑡𝑖+1𝑡𝑖𝜎(𝑡𝑖)=𝑗), and furthermore, 𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎max(𝑡𝑖+1𝑡𝑖𝜎(𝑡𝑖)𝑗,𝜎(𝑡𝑖+1)𝑗) is sufficiently large compared to 𝐾0𝑗(forall𝑗𝑁), according to the constants 𝐴0𝑘(forall𝑘𝑁), the absolute values of the stability abscissas of 𝐴𝑖𝑗(forall𝑖,𝑗𝑁), and norms of 𝐴0𝑗. If there is only a stable matrix 𝐀0 in the set 𝐴0𝑗. If there is a unique stable matrix 𝑗𝑁, for some 𝑡𝑘,𝑡𝑘+1𝑆𝑇𝜎(𝑡𝑘+1𝑡𝑘𝜎(𝑡𝑘)=𝑗), then the switched system is globally exponentially stable only if the switching function is such that 𝐴0𝑗 has infinite measure. If there is a unique stable matrix 𝑗𝑁for some 𝑆𝑇𝜎 and if the sequence of switching instants 𝜎(𝑡𝑘)=𝑗 is finite, then the switching function is such that 𝑡𝑘 for the last switching instant 𝐀0=𝐀0(𝐀0+𝐀0±).
(iv) If 𝐀0 where 𝐀0+, 𝐀0± and 𝐀0 are the sets of stable, unstable, and critically stable matrices in the set 𝜎𝐑0+𝑁 then the switched system is globally exponentially stable independent of the sizes of the delays if the switching function 𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎(𝑡𝑖+1𝑡𝑖𝜎(𝑡𝑖)=𝑗,𝐴0𝑗𝐀0) is such that 𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎(𝑡𝑖+1𝑡𝑖𝜎(𝑡𝑖)=𝑗,𝜎(𝑡𝑖+1)=𝑘,𝐴0𝑗,𝐴0𝑘𝐀0) is sufficiently large compared to 𝐾0𝑗(forall𝑗𝑁) according to the constants 𝐴0𝑘(forall𝑘𝑁), the absolute values of the stability abscissas of 𝐴𝑖𝑗(forall𝑖,𝑗N) and norms of 𝑥(𝑡)2𝑡𝑖ST𝜎(𝑡)𝐾𝑡0𝜎𝑖𝑒1+𝜌𝑡𝑖0𝜎1𝜌𝑡0𝜎𝑖𝑞𝑖=1𝐴𝑡𝑖𝜎𝑖20𝑥0200𝑑×𝑒(𝜌𝑖)0𝜎(𝑡𝐾𝑖)0𝜎(𝑡𝑞𝑖=1||𝐴𝑖)𝑖𝜎(𝑡||2)𝑡𝑖+1𝑡𝑖sup𝜏0𝜑(𝜏)2,(2.18).

Proof. (i) One gets from (2.7) by using Gronwall's lemma [29] then property (i) follows by simple inspection that it is guaranteed that 𝑡 as 𝑥(𝑡)2 since the function of initial condition is bounded on its definition domain.
(ii) It follows directly from the above formula since the upper-bounding function of (𝜌0𝑗)<0is of exponential order with decay rate 𝜌0𝑗>𝐾0𝜎(𝑡𝑖)𝑞𝑖=1𝐴𝑖𝑗2,forall𝑗𝑁, provided thatmax𝑡𝑖ST𝜎(𝑡𝑖+1𝑡𝑖)𝑇, provided that the minimum residence time 𝐀0 is sufficiently large. Properties (iii) and (iv) are direct extensions of Property (ii) for the cases when only one delay-free matrix of dynamics is stable or when only a nonempty subset of them are stable matrices, respectively.

Theorem 2.1 extends known previous ones concerning asymptotic stability of the switched system if all the matrices of the set 𝐀0 are stable and the switching function is subject to a sufficiently large residence time in-between any two consecutive switches. A dual result to Theorem 2.1(i)–(iii) is Theorem 2.2 below for instability when all the matrices in the set 𝐴0𝑗,forall𝑗𝑁 are unstable with no stable or critically stable eigenvalues (i.e., all the matrices (𝐴0𝑗),forall𝑗𝑁, are antistable) and the absolute convergence abscissas of 𝜎𝐑0+𝑁, are sufficiently large compared to the norms of the matrices of delayed dynamics. Note that although the matrices of delay-free dynamics be antistable, any of the parameterizations of the whole delayed system (2.1), (2.3) can be antistable since it is well known that any time invariant delayed system possessing a principal term in its characteristic polynomial has any unstable value at finite distance and there exists only a finite number of modes within each vertical strip. As a result, the number of unstable eigenvalues is finite, and since the system possesses infinitely many eigenvalues [24], one concludes that the system cannot be antistable.

Theorem 2.2. The following properties hold.
(i) The unforced dynamic system (2.1), (2.3) is globally unstable independent of the sizes of the delays if the switching function lim𝑡𝑡𝑖𝑆𝑇𝜎(𝑡)||||𝐾𝑡0𝜎𝑖K𝑡0𝜎𝑖𝑒𝜌𝑖)0𝜎(𝑡1𝜌𝑡0𝜎𝑖||||𝑞𝑖=1A𝑖𝜎(𝑡𝑖)2×𝑒(|̃𝜌𝑖)0𝜎(𝑡|𝐾𝑖)0𝜎(𝑡𝑞𝑖=1||𝐴𝑖)𝑖𝜎(𝑡||2)𝑡𝑖+1𝑡𝑖=,(2.19) is such that where 𝐑϶̃𝜌0𝜎(𝑡𝑖){̃𝜌01,̃𝜌02,,̃𝜌0𝑁} and |̃𝜌0𝑗||𝜌0𝑗|, with 𝐾0𝑗𝐾0𝑗 (with ̃𝜌0𝑗 and 𝐴0𝑗 being located or close to the minimum real part of the eigenvalues of 𝜌0𝑗 and 𝑗𝑁, for all 𝜎(𝑡𝑖)=𝑗𝑁, defined in Theorem 2.1) if 𝑒𝐴0𝑖𝑡2𝐾0𝑖𝑒|̃𝜌0i|𝑡,forall𝑖N are real constants such that 𝐀0 (i.e., all the matrices in the set 𝑆𝑇𝜎(𝑡)={𝑡𝑖𝑆𝑇𝜎𝑡𝑖𝑡} are antistable and then unstable) with 𝑡𝑠(𝑡)+1=𝑡 and 𝑡𝑆𝑇𝜎 if 𝑠(𝑡)=card𝑆𝑇𝜎(𝑡), where 𝜎𝐑0+𝑁.
(ii) The unforced dynamic system (2.1), (2.3) is globally exponentially unstable independent of the sizes of the delays if the switching function 𝐴0𝑗 is such that |̃𝜌0𝑗|>𝐾0𝜎(𝑡𝑖)𝑞𝑖=1𝐴𝑖𝑗2forall𝑗𝑁are all unstable matrices satisfying max𝑡𝑖𝑆𝑇𝜎(𝑡𝑖+1𝑡𝑖)𝑇, and the residence time at each switching instant satisfies 𝑇 with its lower-bound |𝜌0𝑗| being sufficiently large according to the respective absolute values (𝐴0𝑗)of the stability abscissas of the stable matrices . (i.e|𝜌0𝑗|<0, 𝐴0𝑗 if all the eigenvalues of |̃𝜌0𝑗|+𝜀|𝜌0𝑗|+𝜀,𝜀0+ are distinct of positive real parts and 𝑗𝑁, otherwise), for all 𝐴𝑖𝑗(forall𝑖,𝑗𝑁) and norms of 𝜎𝐑0+𝑁.
(iii) The unforced dynamic system (2.1), (2.3) is globally exponentially unstable independent of the sizes of the delays if the switching function 𝐴0𝑗 is such that at least one |𝜌0𝑗|>𝐾0𝜎(𝑡𝑖)𝑞𝑖=1𝐴𝑖𝑗2is a stable matrix satisfyingmax𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎(𝑡𝑖+1𝑡𝑖𝜎(𝑡𝑖)=𝑗), and furthermore, 𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎max(𝑡𝑖+1𝑡𝑖𝜎(𝑡𝑖)𝑗,𝜎(𝑡𝑖+1)𝑗) is sufficiently large compared to 𝐾0𝑗(forall𝑗𝑁), according to the constants 𝐴0𝑘(forall𝑘𝑁), the absolute values of the stability abscissas of 𝐴𝑖𝑗(forall𝑖,𝑗𝑁) and norms of 𝐴0𝑗. If there is only a stable matrix 𝐀0in the set 𝐴0𝑗. If there is a unique stable matrix 𝑗𝑁, for some tk,𝑡𝑘+1𝑆𝑇𝜎(𝑡𝑘+1𝑡𝑘𝜎(𝑡𝑘)=j), then the switched system is globally exponentially stable only if the switching function is such that 𝐴0𝑗 has infinite measure. If there is a unique stable matrix 𝑗𝑁for some 𝑆𝑇𝜎 and if the sequence of switching instants 𝜎(𝑡𝑘)=𝑗 is finite, then the switched system is globally exponentially stable if the switching function is such that 𝑡𝑘 for the last switching instant 𝐋𝑛2.

A combination of Theorems 2.1 and 2.2 will be used in Section 3 to guarantee the boundedness of the input-state and the input-output operators of the switched system. The following result is direct from the fact that if the system is exponentially stable then its Euclidean norm possesses an upper bound of exponential order with negative decay rate so that the state and output trajectory solutions are in 𝐋𝑝2 and Γ, respectively. As a result, the input-state Γ0 and input-output 𝐋(𝐋𝑚2,𝐋𝑛2) operators are members of 𝐋(𝐋𝑚2,𝐋𝑝2) and 𝜎𝐑0+𝑁, respectively, that is, linear and then bounded.

Proposition 2.3. If any of the properties of Theorem 2.1(i)–(iii) hold for a given switching function (Φ𝑥(𝛼))𝐋𝑛2[𝛼,) then the unforced state and output trajectory solutions (𝐶Φ𝑥(𝛼))𝐋𝑝2[𝛼,) and 𝛼𝐑0+, for all Φ𝐋(𝐑𝑛×[,0],𝐋n2[𝛼,)), respectively. Thus, (𝐶Φ)𝐋(𝐑n×[,0],𝐋p2[𝛼,)) and Φ𝐋(𝐑n×[,0],𝐋n2) which are then linear bounded operators since the switched system is either globally asymptotically stable or globally exponentially stable. In particular, (𝐶Φ)𝐋(𝐑n×[,0],𝐋p2) and 𝑢𝐋𝑚2[𝛼,).
If, in addition, 𝛼𝐑 for some (Γ𝑢𝛼)𝐋𝑛2[𝛼,) then the respective forced solutions fulfil (Γ𝑜𝑢)𝐋𝑝2[𝛼,) and 𝛼𝐑0+, for all Γ𝐋(𝐋𝑚2[𝛼,),𝐋𝑛2[𝛼,)), which are then bounded operators. Thus, Γo𝐋(𝐋𝑚2[𝛼,),𝐋𝑝2[𝛼,)) and 𝑢𝐋𝑚2+.
If (Γ+𝑢)𝐋𝑛2+ then the respective forced solutions fulfil (Γ𝑜+𝑢)𝐋𝑝2+ and Γ+𝐋(𝐋𝑚2+,𝐋𝑛2+), respectively, so that Γo+𝐋(𝐋𝑚2+,𝐋𝑝2+) and 𝑢𝐋𝑚2𝐵𝑃𝐶(0)(𝐑0+,𝐑𝑚). Equivalently, if 𝑢(𝑡)=0, that is, 𝑡𝐑, for all Γ+𝐋(𝐋𝑚2+,𝐋𝑛2+), then Γo+𝐋(𝐋𝑚2+,𝐋p2+) and 𝑢𝐋𝑚2𝐵𝑃𝐶(0)(𝐑0+,𝐑𝑚). Equivalently, if Γ𝐋(𝐋𝑚2,𝐋𝑛2), then Γo+𝐋(𝐋𝑚2,𝐋𝑝2) and 𝑢𝐋𝑚2.
If Theorem 2.2 holds and (Γ𝑢)𝐋𝑛2, then the respective forced solutions fulfil (Γo𝑢)𝐋𝑝2 and Γ𝐋(𝐋𝑚2,𝐋𝑛2)so that Γo𝐋(𝐋𝑚2,𝐋𝑝2)and 𝑢𝐋𝑚2𝐵𝑃𝐶(0)(𝐑0,𝐑𝑚). Equivalently, if Γ𝐋(𝐋𝑚2,𝐋𝑛2), then Γo+𝐋(𝐋𝑚2,𝐋𝑝2) and 𝑢.
Proof. The first part concerning the unforced solution follows directly from Theorem 2.1(i)–(iii). The respective linear operators are bounded. The second part follows by taking into account the above properties in Theorem 2.1 and the square-integrability of 𝐋𝑚2 on its appropriate definition domains.
If the system is globally asymptotically stable, then it is possible to restrict the domain and image 𝐋𝑛2 and Γ+ of 𝐋𝑚2+ to 𝐋𝑛2+ and 𝑢𝐋𝑚2+, respectively, for vector functions (Γ+𝑢)𝐋n2+ such that 𝐑0+, since their support is (Γ𝑢)(𝑡)=0,forall𝑡𝐑0 and Γ+𝐋(𝐋m2+,𝐋n2+), and then to define a restricted operator (𝐶Γ+𝐷)+𝐋(𝐋𝑚2+,𝐋𝑝2+). In the same way, it is possible to define a restricted operator Γ𝐋(𝐋𝑚2,𝐋𝑛2). Similarly, it is possible to define (𝐶Γ+𝐷)𝐋(𝐋𝑚2,𝐋𝑝2) and 𝑢𝐋𝑚2 of usefulness for vector functions 𝐑𝑦𝐑± if the system is unstable.

3. Input-State and Input-to-Output Operators of the Switched System and Hankel and Toeplitz Operators Γ[0,𝑡]𝐑𝑚×[0,t]𝐑𝑛×[0,𝑡]

This section investigates the input-state and input-output operators Γ𝑜[0,𝑡]𝐑𝑚×[0,t]𝐑𝑝×[0,t] and 𝑚of the switched system (2.1), and explicit expressions defining them are given. Then, if the input is a square-integrable real 𝐑-vector on Γ𝐋(𝐿𝑚2,𝐿𝑛2), further conditions for Γ𝑜𝐋(𝐿𝑚2,𝐿𝑝2) and Γ𝐑𝑦𝐋(𝐿𝑚2[𝐑𝑦],𝐿𝑛2[𝐑𝑦]) are investigated and weaker ones are also given for Γ𝑜𝑅𝑦𝐋(𝐿𝑚2[𝐑𝑦],𝐿𝑝2[𝐑𝑦]) or 𝐑𝑦𝐑 with 𝐑𝑦𝐑± being a bounded real interval, in particular for 𝐑𝑦𝐑±. Finally, The Hankel and Toeplitz causal and anticausal operators are investigated concerning the cases 𝑢𝐋𝑚2𝐵𝑃𝐶(0)(𝐑,𝐑𝑚). Two different sets of assumptions, the first one being less restrictive, are now given to be used when deriving some of the results of this section.

Assumption 3.1. 𝐴0𝑗, and the matrices 𝐴0𝑗 are dichotomic (i.e., they have no eigenvalues on the imaginary axis) while they have stable and antistable diagonal blocks 𝐴+0𝑗 and 𝑛 of the same respective orders 𝑛+ and 𝑗𝑁, for all  min𝑗𝑁(|𝜌0𝑗|,|̃𝜌0𝑗|)𝜀𝐑+, which satisfy 𝜀. Furthermore, the norms of all the matrices of delayed dynamics are less than 𝐀0 so that Theorem 2.1 (resp., Theorem 2.2) holds if all the matrices in the set 𝐴0𝑗 are stable (resp., antistable).

Assumption 3.2. Assumption 3.1 holds and, furthermore, the matrices 𝑗𝑁 are simultaneously block diagonalizable through the same transformation matrix; for all (𝜀,𝜀)×𝐑.

Note that if Assumption 3.1 hold then no configuration of the switched system has eigenvalues within the open vertical strip 𝑇𝑗 of the complex plane from Theorems 2.1 and 2.2. Furthermore, there exist nonunique coordinate transformations 𝑗𝑁, for all 𝐴0𝑗=𝑇𝑗𝐴0𝑗𝑇𝑗1𝐴=BlockDiag0𝑗,𝐴+0𝑗,(3.1), such that

where 𝐂0 is stable (i.e., all its eigenvalues are in 𝑛) and of order 𝐴+0𝑗, and 𝐂0+ is antistable (i.e., all its eigenvalues are in 𝑛+=𝑛𝑛) and of order 𝑗𝑁, for all 𝑇𝑗=𝑇. Note also that if Assumption 3.2 holds, then 𝑗𝑁, for all ̃𝑥(𝑡)=𝑇(𝑡𝑘)𝑥(𝑡). After a linear change of variables 𝑡[𝑡𝑘,𝑡𝑘+1), for all 𝑡𝑘,𝑡𝑘+1𝑆𝑇𝜎 with 𝜎(𝑡𝑘)=𝑗, such that 𝑇(𝑡𝑘)=𝑇𝑗 and 𝑗𝑁, for some ̇̃𝑥(𝑡)=𝑞𝑖=0𝐴𝑖(𝑡)̃𝑥𝑡𝑖+𝐵(𝑡)𝑢(𝑡),𝑦(𝑡)=𝐶(t)̃𝑥(t)+𝐷(𝑡)𝑢(𝑡),(3.2), the system (2.1) may be described as follows: for all 𝐴𝑖𝐴(𝑡)=𝑖𝑘𝐀𝑖𝐴=𝑖𝑗𝐑𝑛×𝑛𝑗𝑁,𝐵𝐵(𝑡)=𝑘𝐵𝐁=𝑗𝐑𝑛×𝑚𝑗𝑁,𝐶𝐶(𝑡)=𝑘𝐶𝐂=𝑗𝐑𝑝×𝑛𝑗𝑁,𝐷𝐷(𝑡)=𝑘𝐷𝐃=𝑗𝐑𝑝×𝑚𝑗𝑁,(3.3), where

for some 𝜎(𝑡)=𝑘 such that 𝐴𝑖𝐴(𝑡)=𝑖𝑗=𝑇𝑗𝐴𝑖𝑗𝑇𝑗1=𝐴𝑖𝑗𝐴+𝑖𝑗𝐴+𝑖𝑗𝐴++𝑖𝑗,𝑖𝑞{0},andsome𝑗𝐵𝑁,(3.4)𝐵(𝑡)=𝑗=𝑇𝑗𝐵𝑗=𝐵𝑗𝐵+𝑗=𝐶𝐶(𝑡)=𝑗=𝑇𝑗𝐶𝑗𝑇𝑗1𝐶=BlockDiag𝑗,𝐶+𝑗,𝐷𝐷(𝑡)=𝑗=𝐷𝑗,(3.5) subject to (3.1) and (3.4) below: for all 𝐴0𝑗𝐑𝑛×𝑛,𝐴0𝑗𝐑𝑛+×𝑛+,𝐴𝑖𝑗𝐑𝑛×𝑛,𝐴+𝑖𝑗𝐑𝑛×𝑛+,𝐴+𝑖𝑗𝐑𝑛+×𝑛,𝐴++𝑖𝑗𝐑𝑛+×𝑛+,𝐵𝑗𝐑𝑛×𝑚,𝐵+𝑗𝐑𝑛+×𝑚,𝐶𝑗𝐑𝑝×𝑛,𝐶+𝑗𝐑𝑝×𝑛+,𝑖𝑞{0},𝑗𝑁.(3.6) and some for all 𝜒(𝐴0𝑗)=Im𝑇𝑗1[𝐼𝑛0] where

The subspaces 𝑇𝑗 and 𝐴0𝑗 are independent of 𝑗𝑁 and are called, respectively, the stable and antistable subspaces of 𝐑𝑛=𝜒(𝐴0𝑗)𝜒+(𝐴0𝑗), for all 𝑗𝑁, which are complementary, that is, 𝐑𝑛=𝜒(𝐴0(𝑡))𝜒+(𝐴0(𝑡)), for all 𝑡𝐑+0, so that Π𝑗=𝑇𝑗1𝐼𝑛0𝑇00𝑗,Π+𝑗=𝑇𝑗1000𝐼𝑛+𝑇𝑗,𝑗Π𝑁,(t)=𝑇1𝐼(𝑡)𝑛000𝑇(𝑡)=𝑇𝑗1𝐼𝑛0𝑇00𝑗,Π+(𝑡)=𝑇1𝐼(𝑡)𝑛000𝑇(𝑡)=𝑇𝑗1𝐼𝑛0𝑇00𝑗,(3.7), for all Π±𝑗𝑘=Π±(𝑡)=Π±(𝑡𝑘). The projections on those subspaces are given by the respective formulas:

and 𝜎(𝑡)=𝑗𝑘 for somet[𝑡𝑘,𝑡𝑘+1) such that 𝑡𝑘,𝑡𝑘+1ST(𝜎); for all 𝛼=0 for each ̇̃x(t)=𝑇(𝑡)𝑥(𝑡)=𝑍(𝑡,0)𝑇(0)𝑥(0)+𝑞𝑖=00𝑖𝑍(𝑡,𝜏)𝑇(𝜏)𝑥(𝜏)𝑑𝜏+𝑡𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏,(3.8). Thus, from (2.13)–(2.15), and ̃𝑥=𝜑=𝑇(0)𝜑BPC(0)([,0],𝐑𝑛), one gets directly

with initial conditions 𝑍(𝑡,0)𝐶(0)(𝐑0+,𝐑𝑛×𝑛), so that 𝐑0+, SI𝜎 is an everywhere differentiable matrix function on ̇𝑡𝑍(𝑡)=T𝑘̇𝑍(𝑡)=𝑞𝑖=0𝐴𝑖𝑡𝑘𝑇𝑡𝑖𝑍𝑡𝑖𝑡,0,𝑡𝑘,𝑡𝑘+1,𝑡𝑘ST(𝜎),(3.9), with almost everywhere continuous time-derivative except at time instants in 𝐑+, which satisfies:

on 𝑡[𝑡𝑘,𝑡𝑘+1), since 𝑍(t,0)=0, for all 𝑡𝐑, whose unique solution satisfies 𝐼𝑍(𝑡,0)=𝑇(𝑡)𝑍(𝑡,0)=Φ(𝑡,0)𝑛+𝑞𝑖=1𝑡0𝐴Φ(𝑡,𝜏)𝑖(𝜏)𝑇𝜏𝑖𝑍𝜏i,0𝑑𝜏,𝑡𝐑0+.(3.10), for all 𝑒𝐴0(𝑡)𝑡=𝑇(𝑡)𝑒𝐴0(𝑡)𝑡𝑇1𝑒𝐴(𝑡)=BlockDiag0(𝑡)𝑡𝐴,𝑒+0(𝑡)𝑡,(3.11a), and is defined by

Then, so that

for all 𝑡𝑘,𝑡𝑘+1ST𝜎, with 𝑒𝐴0𝑗𝑡=𝑇𝑗𝑒𝐴0𝑗𝑡𝑇𝑗1𝑒𝐴=BlockDiag0𝑗𝑡𝐴,𝑒+0𝑗𝑡,Φ𝑡,𝑡𝑘=𝑇𝑗Φ𝑡,𝑡𝑘𝑇𝑗1;𝑍𝑡,𝑡𝑘=𝑇𝑗𝑍𝑡,𝑡𝑘𝑇𝑗1,(3.12), for all 𝑡[𝑡𝑘,𝑡𝑘+1), for all 𝑡𝑘ST(𝜎) since

for all 𝜎𝐑0+𝑁, for all 𝑇𝜎(t) provided that (𝚪𝑢)(𝑡)=𝑡=𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑡Φ(𝑡,𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖,𝜏𝐵(𝜏)𝑈(𝜏)𝑈𝛾𝑖=𝑑𝛾𝑢(𝜏)𝑑𝜏(3.13a)𝑡𝑍(𝑡,𝜏)Π(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑡𝑍(𝑡,𝜏)Π+=(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏(3.13b)𝑡ΦΠ(𝑡,𝜏)(𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖Π,𝜏×𝑈(𝜏)𝐵(𝜏)(𝜏)𝑈𝛾𝑖𝑢𝑑𝛾(𝜏)𝑑𝜏𝑡ΠΦ(𝑡,𝜏)+(𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖Π,𝜏+×(𝜏)𝐵(𝜏)𝑈(𝜏)𝑈𝛾i𝑑𝛾𝑢(𝜏)𝑑𝜏,(3.13c), and the transformations also apply on the evolution operators when performing the change of variables.

The input-state and input-output operators obtained in (2.13), (2.16), and (2.15), by taking into account (3.5), are now defined explicitly in the subsequent result for a switching function 𝚪𝑜𝑢(𝑡)=𝑡=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)𝑡𝐶(𝑡)Φ(𝑡,𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖×,𝜏𝐵(𝜏)𝑈(𝜏)𝑈𝛾𝑖=𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)(3.14a)𝑡𝐶(𝑡)𝑍(𝑡,𝜏)Π(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑡𝐶(𝑡)𝑍(𝑡,𝜏)Π+=(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)(3.14b)𝑡Π𝐶(𝑡)Φ(𝑡,𝜏)(𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖Π,𝜏(𝜏)𝐵𝜎×(𝜏)𝑈(𝜏)𝑈𝛾𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏𝑡Π𝐶(𝑡)Φ(𝑡,𝜏)+(𝜏)𝐵(𝜏)+𝑞𝑖=1𝑡Φ(0,𝛾)𝐴𝑖(𝛾)𝑍𝛾𝑖Π,𝜏+×(𝜏)𝐵(𝜏)𝑈(𝜏)𝑈𝛾𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡).(3.14c). Note that the input-state operator depends on the state variable transformations while the input-output operator does not depend on the state variables, that is, it does not depend on the matrices 𝑡Φ(t,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝑞𝑖=1𝜏𝑖𝑡0Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖=,𝛾𝐵(𝛾)𝑢(𝛾)𝑑𝛾𝑑𝜏𝑡+Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑞𝑖=1𝑡𝑡0Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖𝐵𝑈,𝛾(𝛾)(𝛾)𝑈𝜏𝑖𝑢=(𝛾)𝑑𝜏𝑑𝛾𝑡Φ(𝑡,𝜏)𝐵𝜎(𝜏)+(𝜏)𝑢(𝜏)𝑑𝜏𝑡𝑞𝑖=1𝑡0Φ(𝑡,𝜏)𝐴𝑖(𝜏)𝑍𝜏𝑖𝐵𝑈,𝛾(𝜏)(𝛾)𝑈𝜏𝑖𝑢=(𝛾)𝑑𝜏𝑑𝛾𝑡Φ+(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏𝑡𝑞𝑖=1𝑡0Φ(𝑡,𝛾)𝐴𝑖(𝜏)𝑍𝛾𝑖𝐵𝑈,𝜏(𝜏)(𝜏)𝑈𝛾𝑖𝑢(𝜏)𝑑𝛾𝑑𝜏.(3.15).

Lemma 3.3. The input-state and input-output operators have the following pointwise expressions:

Proof. It follows directly since the forced solutions of (2.16)-(2.15) may be recalculated by direct manipulation of the integrals as follows:

Now (3.13c)–(3.14c) are further expanded by using the transformation of state variables and the contribution of each interswitching time intervals. The subsequent auxiliary useful notation convention is used to write the mathematical expressions in a very comprehensive way. It is taken into account that there are no switching instants at negative time, that the current time 𝛾[𝑡𝑗,𝑡𝑗+1) may be or not to be a switching instant and that the transformation of variables are given by a nonsingular matrix 𝜎(𝛾)=𝜎(𝑡𝑗) which takes a finite number of 𝑡0= values and which is constant within the semiopen time interval in-between any two consecutive switching instants:

where 𝑗𝑁0𝐍 is a discrete valued function which takes only a finite number of positive integers according to the switching function used.

Lemma 3.4. The input-state and input-output operators have the following expressions:

Proof. It follows directly from Lemma 3.3 by using (3.5), (3.7), and (3.11b), since (3.10), (3.11a), (3.11b), and (3.11c) hold, where for all [𝜏,𝑡)𝐑0+, 𝑇(𝑡,𝜏)=𝑇𝜎𝜏,𝑡, 𝜎𝜏,𝑡[𝜏,𝑡)𝑁𝜏,𝑡𝑁, ΦΦ(𝑡,𝜏)=BlockDiagΦ(𝑡,𝜏),+(𝑡,𝜏)=𝑇(𝑡,𝜏)Φ(𝑡,𝜏)𝑇1(𝑡,𝜏),(3.19) for 𝜎𝜏,𝑡(=𝜎[𝜏,𝑡))[𝜏,𝑡)(𝐑0+)N𝜏,𝑡𝑁.

Lemmas 3.3 and 3.4 will be then used for the explicit definition of the Hankel and Toeplitz operators of the input-state and input-output operators. The following result is useful as an auxiliary one for a subsequent specification of Lemmas 3.3 and 3.4 either for the general case or for the cases when either Assumptions 3.1 or 3.2 hold.

Lemma 3.5. The following properties hold.
(i) [𝜏,𝑡) dependent on the switching instants Φ(𝑡,𝜏) on Φ+(𝑡,𝜏) (i.e., 𝑛(𝜏,𝑡),𝑛+(𝜏,𝑡) depends on the partial switching function Φ𝜏,𝑡𝑖Φ=BlockDiag𝜏,𝑡𝑖,Φ+𝜏,𝑡𝑖𝑡=𝑇𝑖Φ𝜏,𝑡𝑖𝑇1𝑡𝑖𝑒A=BlockDiag0𝑡𝑖𝜏𝑡𝑖A,𝑒+0𝑡𝑖𝜏𝑡𝑖𝑡,𝜏𝑖,𝑡𝑖+1,𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎,(3.20)) such that which is nonsingular for any finite arguments irrespective of Assumption 3.1, where 𝑛+ is the partial switching function with its domain restricted to 𝑇𝜎(𝑡)=𝑇. 𝑡𝐑0+ and 𝐀0 are, in general, of time interval-dependent sizes 𝜎𝑡𝜎0,𝑡(=𝜎[0,𝑡))[0,𝑡)𝑁𝑡𝑁, respectively.
(ii) If Assumption 3.1 holds, then with the first and second square matrix function blocks being convergent and divergent, respectively, and of associate time invariant sizes ΦΦ(𝑡,𝜏)=BlockDiagΦ(𝑡,𝜏),+(𝑡,𝜏)=𝑇Φ(𝑡,𝜏)𝑇1𝑒(3.21)=BlockDiag𝑘(𝑡)𝑖=1A0𝑡𝑖1𝑡𝑖𝑡𝑖1,𝑒𝑘(𝑡)𝑖=1A+0𝑡𝑖1𝑡𝑖𝑡𝑖1,(3.22), 𝑡,𝜏𝐑0+.
(iii) If Assumption 3.2 holds, then (ii) holds with constant 𝑡𝑖,𝑡𝑖+1𝑆𝑇𝜎𝑡, for all Φ±(𝐴𝑡,𝜏)Φ(0,𝛾)𝑖(𝑍𝛾)𝛾𝑖,𝜏𝐵(𝜏)±=Φ±(𝐴𝑡,𝛾)𝑖(𝑍𝛾)𝛾𝑖,𝜏𝐵(𝜏)±,(3.23).
(iv) If Assumption 3.2 holds, and all the matrices in the set 𝐀𝑖(forall𝑖𝑞) defining the switched system by the partial switching function up to time t defined as 𝑡 commute, so that 𝑛, then for all 𝑛, for all 𝑛+. Furthermore, in (3.16) and (3.17) subject to (3.22).
(v) If both assumptions of Property (vi) hold and all the matrices in the set 𝐴Φ(𝑡,𝜏)𝑖(𝑍𝜏)𝛾𝑖,𝜏𝐵(𝜏)±=Φ±(𝐴𝑡,𝜏)±𝑖(𝑍𝜏)±𝛾𝑖𝐵,𝜏±(𝜏);𝑡,𝜏𝐑,(3.24) defining the switched system by the partial switching function up to time Φ(𝜏,𝑡𝑖) have a block diagonal structure with two block matrices of common sizes 𝐴0𝜎(𝑡𝑖) and 𝐴0(𝑡𝑖), then A0(𝑡𝑖)(𝜏𝑡𝑖A)(+0(𝑡𝑖)(𝜏𝑡𝑖)) is block diagonalizable with two nonzero square matrix blocks of time invariant sizes Φ(𝜏,𝑡𝑖Φ)(+(𝜏,𝑡𝑖)) and 𝛼, for all 𝛽. Furthermore, in (3.16) and (3.17).

Proof. (i) It follows directly from the fact that any real matrix has a Jordan diagonal form.
(ii), (iii) They follow directly from the fact that the matrix function 𝑒𝐴10𝑗𝛼𝑒𝐴20𝑗𝛽=𝑒(𝐴0j1𝛼+𝐴20𝑗𝛽) is an exponential matrix function of 𝑗1,2𝑁𝑡 within interswitching time intervals which is block diagonalizable under the same similarity transformation and with the same block diagonal matrices sizes as the matrixΦ(𝑡,𝜏), the stable (antistable) block diagonal matrix 𝑛 generating a convergent (divergent) exponential matrix function 𝑛+.
(iv) Its first part follows from (2.9) since for any real constants 𝐀𝑖, 𝑖𝑞 and any 𝐀0 which commute, 𝐏𝑟±𝐋𝑟2𝐋𝑟2±, for all 𝐋𝑟2. Its second part follows from the semigroup property of 𝐋𝑟2±.
(v) It follows from (2.9) and (2.15), both being block diagonal with two non-zero square block matrices of corresponding identical time invariant sizes, respectively, Γ+𝐋𝑚2+𝐋𝑛2+ and Γ0+𝐋𝑚2+𝐋𝑝2+, under the given assumptions since the matrices Γ+𝐋𝑚2𝐋𝑛2, for all Γ0+𝐋𝑚2𝐋𝑝2, are diagonalizable with identical two square matrix blocks of identical sizes.

If all the matrices in the set 𝑡𝐑 are dichotomic, namely, they have no critically stable eigenvalues, then they admit a similarity transformation to a block diagonal form with only stable and instable eigenvalues. Under some extra assumptions related to the switching function to require a minimum residence time at each parameterization of the switched system, it may be proved that the input-state/output operators of the solution are bounded operators. Now, denote by 𝑡𝐑 the usual orthogonal projections of 𝐑0+ontoΓ𝐋(𝐋𝑚2,𝐋𝑛2). Those projections are useful to describe the input-state and input-output operators for positive or negative times when the input is least square-integrable either for the negative or positive real semiaxis. The subsequent previous results are direct.

Lemma 3.6. Γ𝑜𝐋(𝐋𝑚2,𝐋𝑝2) and Γ+𝐋𝑚2+𝐋𝑛2+ are linear bounded, equivalently continuous, operators if any of the properties Theorem 2.1(i)–(iii) holds, and Γ𝑜+𝐋𝑚2+𝐋𝑝2+ and 𝐋𝑚2 are linear bounded, equivalently continuous, operators if any of the properties in Theorem 2.2 holds.

Proof. It turns out from applying the Cauchy-Schwartz inequality to the sate/output-trajectory solutions that if the system is globally asymptotically stable and the input is an original (i.e., it is identically zero for 𝐋𝑚2+) and, furthermore, square-integrable, then the state and output trajectory solutions are identically zero for 𝐋𝑛2 and square-integrable on 𝐋𝑝2. As a result, both linear operators are bounded and, equivalently, continuous. The second result is a dual one to the first result.

Note that, compared to 𝐋𝑛2+ and 𝐋𝑝2+, the input-state operators Γ𝐋(𝐋𝑚2,𝐋𝑛2) and input-output Γ𝑜𝐋(𝐋𝑚2,𝐋𝑝2) (identified with the so-called causal Toeplitz operator if the input is an original vector function) have domains restricted from Γ𝐋𝑚2𝐋𝑛2 toΓ𝑜𝐋𝑚2𝐋𝑝2 and projected images from 𝐋𝑚2, respectively, 𝐋𝑚2, onto 𝐋𝑛2, respectively,𝐋𝑝2, provided that 𝐋𝑛2+ and 𝐋𝑝2+. In the same way, the input-state operators Γ𝐋(𝐋𝑚2,𝐋𝑛2) and input-output Γ𝑜𝐋(𝐋𝑚2,𝐋𝑝2) have domains restricted from Γ𝐋(𝐋𝑚2,𝐋𝑛2) to Γ𝑜𝐋(𝐋𝑚2,𝐋𝑝2) and projected images from 𝐋𝑚,𝑛𝑝2±, respectively, Γ𝐋(𝐋𝑚2,𝐋𝑛2), onto Γ𝑜𝐋(𝐋𝑚2,𝐋𝑝2), respectively, 𝐇Γ𝑜𝐏𝑝+Γ𝑜|𝐋𝑚2=𝐏𝑝+Γ𝑜𝐏𝑚, provided that Γ𝑜 and 𝐇Γ𝑜𝐏𝑝Γ𝑜|𝐋𝑚2+=𝐏𝑝Γ𝑜𝐏𝑚+. Note also that Lemma 3.6 only gives sufficiency-type conditions of boundedness of those operators based on results of Theorems 2.1, 2.2. The following definitions are related to four important input-to-sate and input-output operators which are obtained from the operators Γ𝑜 and 𝐓Γ𝑜𝐏𝑝+Γ𝑜|𝐋𝑚2+=𝐏𝑝+Γ𝑜𝐏𝑚+ subject to domain restrictions and orthogonal projections of their images since they act on half axis Lebesgue spaces Γ𝑜.

Definition 3.7. Let 𝐓Γ𝑜𝐏𝑝Γ𝑜|𝐋𝑚2=𝐏𝑝Γ𝑜𝐏𝑚 be bounded, so that Γ𝑜 is also bounded. We define the following:(1)the causal input-output Hankel operator (or, simply causal Hankel operator) 𝐇Γ𝐏𝑛+Γ|𝐋𝑚2=𝐏𝑛+Γ𝐏𝑚 with symbol Γ,(2)the anticausal input-output Hankel operator (or, simply anticausal Hankel operator) 𝐇Γ𝐏𝑛Γ|𝐋𝑚2+=𝐏𝑛Γ𝐏𝑚+ with symbol Γ,(3)the causal input-output Toeplitz operator (or, simply causal Toeplitz operator) 𝐓Γ𝐏𝑛+Γ|𝐋𝑚2+=𝐏𝑛+Γ𝐏𝑚+ with symbol Γ,(4)the anticausal input-output Toeplitz operator (or, simply anticausal Toeplitz operator) 𝐓Γ𝐏𝑛Γ|𝐋𝑚2=𝐏𝑛Γ𝐏𝑚 with symbol Γ,(5)the causal input-state Hankel operator 𝐓Γ𝑜+𝐇Γ𝑜=Γ𝑜𝐏𝑚+ with symbol 𝐓Γ𝑜=𝐏𝑝+Γ𝑜𝐏𝑚+,(6)the anticausal input-state Hankel operator 𝐇Γ0=𝐏𝑝Γ𝑜𝐏𝑚+=0 with symbol 𝚪𝑜𝐏𝑚+𝑢(𝑡)=𝑡0𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),(3.25),(7)the causal input-state Toeplitz operator 𝐓Γ𝑜+𝐇Γ𝑜=Γ𝑜𝐏𝑚 with symbol 𝐓Γ𝑜=𝐏𝑝Γ𝑜𝐏𝑚,(8)the anticausal input-state Toeplitz operator 𝐇Γ𝑜=𝐏𝑝+Γ𝑜𝐏𝑚=0 with symbol 𝚪𝑜𝐏𝑚𝑢(𝑡)=0𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),(3.26).

The input-output Hankel and Toeplitz operators (see Definitions  3.7 [14]), or simply Hankel and Toeplitz operators, are of wide use for the particular case of delay-free systems with single parameterizations, then being delay-free linear time invariant systems (see, for instance, [25]). Definition 3.7 and Lemmas 3.3, 3.4 define extensions of those operators to describe the input-state/output trajectories of the time delayed switched system (2.1). The subsequent result related to the state and output trajectory solutions of the switched system (2.1) are described by the input-sate and input-output Hankel and Toeplitz operators.

Theorem 3.8. The following properties hold under Assumption 3.1:
(i) 𝐓Γ+𝐇Γ=Γ𝐏𝑚+, so that 𝐓Γ=𝐏𝑝+Γ𝐏𝑚+ if and only if 𝐇Γ=𝐏𝑝Γ𝐏𝑚+=0, with 𝐓Γ+𝐇Γ=Γ𝐏𝑚, so that 𝐓Γ=𝐏𝑝Γ𝐏𝑚 if and only if 𝐇Γ=𝐏𝑝+Γ𝐏𝑚=0, with
(ii) (𝐇Γ0𝑢)(𝑡)=0, so that 𝐓𝚪𝑜𝑢𝐏(𝑡)=𝑝+𝚪𝑜𝐏𝑚𝑢=(𝑡)𝑡0𝐶=(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)𝑡0=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)𝑘(𝑡)𝑗=1𝑡𝑗𝑡𝑗1𝐶𝑡𝑗1𝑡Φ(𝑡,𝜏)𝐵𝑗1+𝑞𝑖=1𝑘(𝑡)=1𝑡𝑡1Φ(𝑡,𝛾)𝐴𝑖𝑡1𝑍𝛾𝑖𝑡,𝜏×𝐵𝑗1𝑈(𝜏)𝑈𝛾𝑖𝑢=𝑑𝛾(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)𝑘(𝑡)𝑗=1𝑡𝑗𝑡𝑗1𝐶𝑡𝑗1𝐵𝑡Φ(𝑡,𝜏)𝑗1+𝑞𝑖=1𝑘(𝑡)=1𝑡𝑡1𝐴Φ(𝑡,𝛾)𝑖𝑡1𝑍𝛾𝑖×𝐵𝑡,𝜏𝑗1𝑈(𝜏)𝑈𝛾𝑖+𝑑𝛾𝑢(𝜏)𝑑𝜏𝐷(𝑡)𝑢(𝑡),𝑡𝐑0+.(3.29) if and only if 𝑡1=0, with 𝑡1>0, so that 𝑆𝑇𝜎 if and only if 𝑡10, with
(iii) 𝑖1, The last expression being valid if 𝑡(,𝑡2] since 𝐓𝚪𝑜𝑢𝐇(𝑡)=0,𝚪𝑜𝑢𝐏(𝑡)=𝑝𝚪𝑜𝐏𝑚+𝑢(𝑡)=0𝐶+(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)+𝑢(𝜏)𝑑𝜏=𝑗=𝑘(𝑡)𝑡𝑗+1𝑡𝑗𝐶+Φ(𝑡)+𝐵(𝑡,𝜏)+𝑡𝑗+𝑞𝑖=1𝑘(𝑡)=1𝑡𝑡1𝐴Φ(𝑡,𝛾)𝑖𝑡1𝑍𝛾𝑖𝐵𝑡,𝜏𝑗+×𝑈(𝜏)𝑈𝛾𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏,𝑡𝐑0,(3.30). If 𝑡1=0, then the given switching sequence 𝜎(𝑡)=𝜎() may be redefined as 𝑡(,𝑡2], (𝐓Γ𝑜𝑢)(𝑡)=0 for all 𝐇𝚪𝑜𝑢𝐏(𝑡)=𝑝+𝚪𝑜𝐏𝑚𝑢=(𝑡)0=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑0=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏