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

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 [3–13]. 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, [6–27]. 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, 12–18]). 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, 19–24, 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 [11–13, 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.
𝐙₋and𝐑₋ 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^(𝑖−1)(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 𝐋^𝑛2≡𝐋^𝑛2(𝐑) and endowed with the inner product 𝐿²-norm ‖𝑓‖_𝐋^𝑛2∶=(∫^∞−∞‖𝑓(𝜏)‖²²𝑑𝜏), for all 𝑓∈𝐋^𝑛2, where ‖⋅‖₂ is the ℓ₂-vector (or Euclidean) norm and its corresponding induced matrix norm. 𝐋^𝑛2[𝛼,∞) the Hilbert space of 𝑛 norm-square Lebesgue integrable real functions on [𝛼,∞)⊂𝐑 for a given 𝛼∈𝐑 which is endowed with the norm ‖𝑓‖_𝐋^𝑛2_[𝛼,∞)∶=(∫^∞𝛼‖𝑓(𝜏)‖²²𝑑𝜏), for all 𝑓∈𝐋^𝑛2[𝛼,∞). 𝐋^𝑛2+∶={𝑓∈𝐋^𝑛2∶𝑓(𝑡)=0,forall𝑡∈𝐑₋} and 𝐋^𝑛2−∶={𝑓∈𝐋^𝑛2∶𝑓(𝑡)=0,forall𝑡∈𝐑₊} are closed subspaces of 𝐋^𝑛𝟐∶={𝑓∈𝐋^𝑛2∶𝑓(𝑡)=0,forall𝑡∈𝐑₋}→𝐋^𝑛𝟐 of respective supports 𝐑₀₊ and 𝐑₀₋. Then, 𝐋^𝑛2=𝐋^𝑛2+⊕𝐋^𝑛2−.

𝐼_𝑛 denotes the 𝑛th identity matrix.

𝜆_max(𝑀) and 𝜆_min(𝑀) stand for the maximum and minimum eigenvalues of a definite square real matrix 𝑀=(𝑚_𝑖𝑗).

𝜎∶𝐑₀₊→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. 𝜎_𝜏,𝑡(∶=𝜎∣[0,t))∶[0,t)(⊂𝐑₀₊)→𝑁_𝜏,𝑡⊂𝑁 is the partial switching function with its domain restricted to [𝜏,𝑡]. 𝜎_𝑡is a notational abbreviation of 𝜎_0,𝑡.

The point constant delays are denoted by ℎ_𝑖∈[0,ℎ],forall𝑖∈𝑞∪{0} and are, in general, incommensurate, and ℎ₀=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=ℎ₀<ℎ₁<ℎ₂<⋯<ℎ_𝑞=ℎ:

where ℎ_𝑖∈(0,∞); for all 𝑖∈𝑞∶={1,2,…,𝑞}, 𝑥(𝑡)∈𝐑^𝑛, 𝑢(𝑡)∈𝐑^𝑚, and 𝑦(𝑡)∈𝐑^𝑝are the state, input (or control) and output (or measurement) vectors, respectively, and

where 𝑖∈𝑞∪{0}∶={0,1,2,…,𝑁}, fulfilling that 𝐴_𝑖(𝜏), 𝐵_𝑖(𝜏), 𝐶_𝑖(𝜏) and 𝐷_𝑖(𝜏) are piecewise constant such that they are constant either in (𝑡−𝑇,𝑡] or in [𝑡,𝑡+𝑇), for all 𝑡∈𝐑₀₊ and some fixed 𝑇∈𝐑₊. 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):𝑥̇(𝑡)=(∑^𝑞𝑖=0𝐴_𝑖(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:  𝑥̇(𝑡)=𝐴₀(𝑡)𝑥(𝑡−ℎ_𝑖);  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 𝜎∶𝐑₀₊→𝑁, which generates a strictly ordered sequence of switching time instants 𝑆𝑇_𝜎∶={𝑡_𝑖∶𝑡_𝑖+1≥𝑡_𝑖+𝑇,forall𝑖∈𝑁₀(⊃{1})⊂𝐙₊,𝑡₁∈𝐑₀₊}, and which might be equivalently rewritten, since 𝐴_𝑖(𝑡)=𝐴_{𝑖𝜎(𝑡)}, for all 𝑖∈𝑁∪{0}, 𝐵(𝑡)=𝐵_𝜎(𝑡), 𝐶(𝑡)=𝐶_𝜎(𝑡), 𝐷(𝑡)=𝐷_𝜎(𝑡) via the switching function 𝜎∶𝐑₀₊→𝑁, as

where 𝐑₀₊ is the state-trajectory solution, which is almost everywhere time differentiable on [−ℎ,0) and satisfies (2.3), subject to bounded piecewise continuous initial conditions on 𝑥=𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛), that is, 𝜎(𝑡)=𝑗∈𝑁,forall𝑡∈𝐑₋∪[0,𝑡₁). It is assumed that 𝑡₁∈ST_𝜎, 𝜎∶𝐑₀₊→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 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) according to Picard-Lindeloff theorem for any given 𝑢∈BPC⁽⁰⁾(𝐑,𝐑^𝑚) and any [𝛼,𝑡]⊂𝐑, may be calculated on any time interval [𝛼,𝑡]=𝛼,𝑡_𝑘∪⎛⎜⎜⎜⎝_𝑖∈𝑁_𝑡(𝛼)𝑡_𝑘+𝑖,𝑡_𝑘+𝑖+1⎞⎟⎟⎟⎠∪𝑡_{𝑘+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 𝑁_𝑡(𝛼)∶={𝑖∈𝐍∶SI_𝜎∋𝑡_𝑖≤𝑡}; 𝑡_𝑘+𝑖∈ST_𝜎, 𝑖∈𝑁_𝑡(𝛼), for all 𝑡_{𝑘+𝑁t(𝜎)+1}∈ST_𝜎 and 𝜎(𝑡^−𝑘)=𝑗(𝑡_𝑘)≠𝜎(𝑡^+𝑘)=𝑗(𝑡_𝑘+1)∈𝑁. Note that 𝑡_𝑘,𝑡_𝑘+1∈ST_𝜎, for all 𝑥(𝑡)==Φ𝚽(𝑡𝑥,_ℎ𝛼()𝛼𝑥)(𝛼(𝑡))++_𝑞𝚪𝑢_𝛼(𝑡)_𝑖=1^𝑡𝛼Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑥𝜏−ℎ_𝑖𝑑𝜏+^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 𝜎_𝜏,𝑡 depends on the corresponding partial switching function Φ(𝑡,𝜏), the simpler notation Φ_{𝜎𝜏,𝑡}(𝑡,𝜏) is preferred instead for 𝑀_𝜎(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)==𝐶𝐶(𝚽𝑡)𝑥_ℎΦ(𝛼(t),𝛼()𝑡)𝑥+(𝛼)𝐶+𝚪𝑢_𝛼𝑞+𝐷(𝑡)_𝑖=1^𝑡𝛼Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑥𝜏−ℎ_𝑖𝑑𝜏+^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 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛), subject to initial conditions 𝑥_ℎ(𝛼), where

(1) [𝛼−ℎ,ℎ] is the strip of state-trajectory solution on 𝜑(𝑡) which takes values 𝑡=𝛼−ℎ<0 if Φ∈𝐋(𝐑^𝑛×𝐑,𝐑^𝑛)

(2) the evolution operator in 𝚽𝑥_ℎ(𝛼)(𝑡)∶=Φ(𝑡,𝛼)𝑥(𝛼)+^𝑞_𝑖=1^𝑡𝛼Φ(𝑡,𝜏)𝑥𝜏−ℎ_𝑖𝑑𝜏,∀𝑡(≥𝛼),𝛼∈𝐑,(2.8) is defined pointwisely by

so that [0,𝑡] is the unforced response in Φ∈𝐶⁽⁰⁾(𝐑×𝐑,𝐑^𝑛×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 ST_𝜎with bounded discontinuities on the set [𝛼,𝑡]⊂𝐑 and is defined on the interval Φ(𝑡,𝛼)=𝑒^{𝐴0(𝑡𝑘+𝑁𝑡(𝛼)+1)(𝑡−𝑡𝑘+𝑁𝑡(𝛼)+1)}⎛⎜⎜⎝^{𝑘+𝑁𝑡(𝛼)}_𝑖=1𝑒^{𝐴0𝑡𝑘+𝑖𝑡𝑘+𝑖+1−𝑡𝑘+𝑖}⎞⎟⎟⎠𝑒^{𝐴0𝑡(𝛼)𝑘−𝛼},(2.9) as

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

(3) the input-state and input-output operators in Γ_𝑜∈𝐋(𝐑^𝑚×𝐑,𝐑^𝑝)and Γ_𝑜∶=𝐶_𝜎Γ+𝐷_𝜎, respectively, 𝚪𝚪𝑢_𝛼𝑡(𝑡)∶=^𝑡𝛼Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏=^𝑡−∞Φ(𝑡,𝜏)𝐵(𝜏)𝑢_𝛼𝑡(𝜏)𝑑𝜏_𝑜𝑢_𝛼𝑡(𝑡)∶==^𝑡𝛼𝐶(𝑡)Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)_𝑡−∞𝐶(𝑡)Φ(𝑡,𝜏)𝐵(𝜏)𝑢_𝛼𝑡(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),∀𝑡(≥𝛼),𝛼∈𝐑₀₊,(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)+^𝑞_𝑖=1^{𝛼𝛼−ℎ}_𝑖𝑦(𝑡)=𝐶(𝑡)𝑍(𝑡,𝛼)𝑥(𝛼)+_𝑞𝑍(𝑡,𝜏)𝑥(𝜏)𝑑𝜏+^𝑡−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢_𝛼𝑡(𝜏)𝑑𝜏,_𝑖=1^{𝛼𝛼−ℎ}_𝑖𝑍(𝑡,𝜏)𝑥(𝜏)𝑑𝜏+^𝑡−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢_𝛼𝑡(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,𝐑^𝑛×𝑛), 𝐑₊ is an everywhere differentiable matrix function on SI_𝜎, with almost everywhere continuous time-derivative except at time instants in 𝑍̇(𝑡)=^𝑞_𝑖=0𝐴(𝑡)𝑍𝑡−ℎ_𝑖,0(2.14), which satisfies

on 𝑍(𝑡,𝛼)=0 whose unique solution satisfies 𝛼(<𝑡), for all 𝑡∈𝐑, 𝑍(𝑡,𝛼)=Φ(𝑡,𝛼)𝐼_𝑛+^𝑞_𝑖=1^𝑡𝛼Φ(𝛼,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛼𝑑𝜏,∀𝑡(≥𝛼),𝛼∈𝐑(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 𝑥(𝑡)=Φ(t,0)𝐼_𝑛+^𝑞_𝑖=1^𝑡0+Φ(0,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,0𝑑𝜏𝑥(0)_𝑞_𝑞𝑖=1_𝑗=1^t₀^0−ℎ_𝑗+Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏_𝑡−∞Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+^𝑞_𝑖=1^𝑡0^{𝜏−ℎ𝑖}_−∞=Φ(𝑡,0)𝐼_𝑛Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝐵(𝛾)𝑢(𝛾)𝑑𝛾𝑑𝜏+^𝑞_𝑖=1^𝑡0+Φ(0,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,0𝑑𝜏𝑥(0)_𝑞_𝑞𝑖=1_𝑗=1^𝑡0^0−ℎ_𝑗+Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏_𝑡−∞Φ(𝑡,𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞𝑦(𝑡)=𝐶_𝜎(𝑡)Φ(0,𝛾)𝐴_𝑖(𝛾)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏,Φ(𝑡,0)𝐼^𝑛+^𝑞_𝑖=1^𝑡0+Φ(0,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,0𝑑𝜏𝑥(0)_𝑞_𝑞𝑖=1_𝑗=1^𝑡0^0−ℎ_𝑗+Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏+^𝑡−∞Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_𝑞_𝑖=1^𝑡0^{𝜏−ℎ𝑖}_−∞=𝐶(𝑡)Φ(𝑡,0)𝐼_𝑛Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝐵(𝛾)𝑢(𝛾)𝑑𝛾𝑑𝜏+𝐷(𝑡)𝑢(𝑡)+^𝑞_𝑖=1^𝑡0+Φ(0,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,0𝑑𝜏𝑥(0)_𝑞_𝑞𝑖=1_𝑗=1^𝑡0^0−ℎ_𝑗+Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝜑(𝛾)𝑑𝛾𝑑𝜏_𝑡−∞Φ(𝑡,𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞×𝐵(𝜏)Φ𝑈(0(,𝜏𝛾))−𝐴𝑈_𝑖(𝛾)𝛾𝑍−ℎ𝛾_𝑖−ℎ_𝑖,𝜏𝑑𝛾𝑢(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 𝜎∶𝐑₀₊→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 ∃lim_{𝑡→∞𝑡𝑖∈}_{𝑆𝑇𝜎(𝑡)}0𝑥0200𝑑𝐾_{𝑡0𝜎𝑖}1+𝑒^{𝜌0𝜎𝑡𝑖ℎ}−1𝜌_{𝑡0𝜎𝑖}^𝑞_𝑖=1‖‖𝐴_{𝑡𝑖𝜎𝑖}‖‖₂×𝑒_{−(𝜌𝑖)0𝜎(𝑡−𝐾𝑖)0𝜎(𝑡∑𝑞𝑖=1||𝐴𝑖)𝑖𝜎(𝑡||2)𝑡𝑖+1−𝑡𝑖}=0,(2.17) is such that where 𝐑₊϶𝜌_{0𝜎(𝑡𝑖)}∈{𝜌₀₁,𝜌₀₂,…,𝜌_0𝑁} and 𝜎(𝑡_𝑖)=𝑗∈𝑁 if ‖𝑒^{𝐴0𝑖𝑡}‖₂≤𝐾_0𝑖𝑒^{−𝜌0𝑖𝑡},forall𝑖∈𝑁 are real constants such that 𝐀₀ (i.e., all the matrices in the set 𝑆𝑇_𝜎(𝑡)∶={𝑡_𝑖∈𝑆𝑇_𝜎∶𝑡_𝑖≤𝑡} are stable) with 𝑡_𝑠(𝑡)+1∶=𝑡 and 𝑡∉𝑆𝑇_𝜎 if s(𝑡)∶=card𝑆𝑇_𝜎(𝑡), where 𝜎∶𝐑₀₊→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 𝐴_0𝑗 is such that 𝜌_0𝑗>𝐾_{0𝜎(𝑡𝑖)}∑^𝑞𝑖=1‖𝐴_𝑖𝑗‖₂are all stable matrices satisfying 𝑗∈𝑁, for all max_{𝑡𝑖∈𝑆𝑇𝜎}(𝑡_𝑖+1−𝑡_𝑖)≥𝑇, and the residence time at each switching instant satisfies |𝜌_0𝑗| with its lower-bound T being sufficiently large according to the respective absolute values 𝐴_0𝑗 of the stability (or convergence) abscissas of −𝜌_0𝑗<0 (i.e., 𝐴_0𝑗 if all the eigenvalues of −𝜌_0𝑗+𝜀,𝜀→0⁺ are distinct and 𝑗∈𝑁, otherwise), for all 𝐴_𝑖𝑗(forall𝑖∈𝑞,𝑗∈𝑁) and the norms of the matrices 𝜎∶𝐑₀₊→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 𝐴_0𝑗 is such that at least one 𝜌_0𝑗>𝐾_{0𝜎(𝑡𝑖)}∑^𝑞𝑖=1‖𝐴_𝑖𝑗‖₂is a stable matrix satisfying max_{𝑡𝑖,𝑡𝑖+1∈𝑆𝑇𝜎}(𝑡_𝑖+1−𝑡_𝑖∶𝜎(𝑡_𝑖)=𝑗), and furthermore, ∑_{𝑡𝑖,𝑡𝑖+1∈𝑆𝑇𝜎}max(𝑡_𝑖+1−𝑡_𝑖∶𝜎(𝑡_𝑖)≠𝑗,𝜎(𝑡_𝑖+1)≠𝑗) is sufficiently large compared to 𝐾_0𝑗(forall𝑗∈𝑁), according to the constants 𝐴_0𝑘(forall𝑘∈𝑁), the absolute values of the stability abscissas of 𝐴_𝑖𝑗(forall𝑖,𝑗∈𝑁), and norms of 𝐴_0𝑗. If there is only a stable matrix 𝐀₀ in the set 𝐴_0𝑗. If there is a unique stable matrix 𝑗∈𝑁, for some ∑_{𝑡𝑘,𝑡𝑘+1∈𝑆𝑇𝜎}(𝑡_𝑘+1−𝑡_𝑘∶𝜎(𝑡_𝑘)=𝑗), then the switched system is globally exponentially stable only if the switching function is such that 𝐴_0𝑗 has infinite measure. If there is a unique stable matrix 𝑗∈𝑁for some 𝑆𝑇_𝜎 and if the sequence of switching instants 𝜎(𝑡_𝑘)=𝑗 is finite, then the switching function is such that 𝑡_𝑘 for the last switching instant 𝐀₀=𝐀₀₋∪(𝐀₀₊∪𝐀_0±).
(iv) If 𝐀₀₋≠∅ where 𝐀₀₊, 𝐀_0± and 𝐀₀ are the sets of stable, unstable, and critically stable matrices in the set 𝜎∶𝐑₀₊→𝑁 then the switched system is globally exponentially stable independent of the sizes of the delays if the switching function ∑_{𝑡𝑖,𝑡𝑖+1∈𝑆𝑇𝜎}(𝑡_𝑖+1−𝑡_𝑖∶𝜎(𝑡_𝑖)=𝑗,𝐴_0𝑗∈𝐀₀₋) is such that ∑_{𝑡𝑖,𝑡𝑖+1∈𝑆𝑇𝜎}(𝑡_𝑖+1−𝑡_𝑖∶𝜎(𝑡_𝑖)=𝑗,𝜎(𝑡_𝑖+1)=𝑘,𝐴_0𝑗,𝐴_0𝑘∉𝐀₀₋) is sufficiently large compared to 𝐾_0𝑗(forall𝑗∈𝑁) according to the constants 𝐴_0𝑘(forall𝑘∈𝑁), the absolute values of the stability abscissas of 𝐴_𝑖𝑗(forall𝑖,𝑗∈N) and norms of ‖𝑥(𝑡)‖₂≤_{𝑡𝑖∈ST𝜎(𝑡)}𝐾^{𝑡0𝜎𝑖}1+𝑒^{𝜌𝑡𝑖0𝜎ℎ}−1𝜌_{𝑡0𝜎𝑖}^𝑞_𝑖=1‖‖𝐴_{𝑡𝑖𝜎𝑖}‖‖₂×𝑒_{−(𝜌𝑖)0𝜎(𝑡−𝐾𝑖)0𝜎(𝑡∑𝑞𝑖=1||𝐴𝑖)𝑖𝜎(𝑡||2)𝑡𝑖+1−𝑡𝑖}0𝑥0200𝑑sup_{−ℎ≤𝜏≤0}‖𝜑(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 (−𝜌_0𝑗)<0is of exponential order with decay rate 𝜌_0𝑗>𝐾_{0𝜎(𝑡𝑖)}∑^𝑞𝑖=1‖𝐴_𝑖𝑗‖₂,forall𝑗∈𝑁, provided thatmax_{𝑡𝑖∈ST𝜎}(𝑡_𝑖+1−𝑡_𝑖)≥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 𝐴_0𝑗,forall𝑗∈𝑁 are unstable with no stable or critically stable eigenvalues (i.e., all the matrices −(𝐴_0𝑗),forall𝑗∈𝑁, are antistable) and the absolute convergence abscissas of 𝜎∶𝐑₀₊→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 ∃lim_{𝑡→∞𝑡𝑖∈}_{𝑆𝑇𝜎(𝑡)}||||𝐾^{𝑡0𝜎𝑖}−K_{𝑡0𝜎𝑖}𝑒^{𝜌𝑖)0𝜎(𝑡ℎ}−1𝜌_{𝑡0𝜎𝑖}||||^𝑞_𝑖=1‖‖A_{𝑖𝜎(𝑡𝑖)}‖‖₂×𝑒_(|𝜌̃_{𝑖)0𝜎(𝑡|−𝐾𝑖)0𝜎(𝑡∑}^𝑞𝑖=1_{||𝐴𝑖)𝑖𝜎(𝑡||2)𝑡𝑖+1−𝑡𝑖}=∞,(2.19) is such that where 𝐑₋϶𝜌̃_{0𝜎(𝑡𝑖)}∈{𝜌̃₀₁,𝜌̃₀₂,…,𝜌̃_0𝑁} and |𝜌̃_0𝑗|≤|𝜌_0𝑗|, with 𝐾_0𝑗≤𝐾_0𝑗 (with 𝜌̃_0𝑗 and 𝐴_0𝑗 being located or close to the minimum real part of the eigenvalues of 𝜌_0𝑗 and 𝑗∈𝑁, for all 𝜎(𝑡_𝑖)=𝑗∈𝑁, defined in Theorem 2.1) if ‖𝑒^{𝐴0𝑖𝑡}‖₂≥𝐾_0𝑖𝑒^|𝜌̃_0i|𝑡,forall𝑖∈N are real constants such that 𝐀₀ (i.e., all the matrices in the set 𝑆𝑇_𝜎(𝑡)∶={𝑡_𝑖∈𝑆𝑇_𝜎∶𝑡_𝑖≤𝑡} are antistable and then unstable) with 𝑡_𝑠(𝑡)+1∶=𝑡 and 𝑡∉𝑆𝑇_𝜎 if 𝑠(𝑡)∶=card𝑆𝑇_𝜎(𝑡), where 𝜎∶𝐑₀₊→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 𝐴_0𝑗 is such that |𝜌̃_0𝑗|>𝐾_{0𝜎(𝑡𝑖)}∑^𝑞𝑖=1‖𝐴_𝑖𝑗‖₂forall𝑗∈𝑁are all unstable matrices satisfying max_{𝑡𝑖∈𝑆𝑇𝜎}(𝑡_𝑖+1−𝑡_𝑖)≥𝑇, and the residence time at each switching instant satisfies 𝑇 with its lower-bound |𝜌_0𝑗| being sufficiently large according to the respective absolute values (−𝐴_0𝑗)of the stability abscissas of the stable matrices . (i.e−|𝜌_0𝑗|<0, 𝐴_0𝑗 if all the eigenvalues of −|𝜌̃_0𝑗|+𝜀≤−|𝜌_0𝑗|+𝜀,𝜀→0⁺ are distinct of positive real parts and 𝑗∈𝑁, otherwise), for all 𝐴_𝑖𝑗(forall𝑖,𝑗∈𝑁) and norms of 𝜎∶𝐑₀₊→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 𝐴_0𝑗 is such that at least one |𝜌_0𝑗|>𝐾_{0𝜎(𝑡𝑖)}∑^𝑞𝑖=1‖𝐴_𝑖𝑗‖₂is a stable matrix satisfyingmax_{𝑡𝑖,𝑡𝑖+1∈𝑆𝑇𝜎}(𝑡_𝑖+1−𝑡_𝑖∶𝜎(𝑡_𝑖)=𝑗), and furthermore, ∑_{𝑡𝑖,𝑡𝑖+1∈𝑆𝑇𝜎}max(𝑡_𝑖+1−𝑡_𝑖∶𝜎(𝑡_𝑖)≠𝑗,𝜎(𝑡_𝑖+1)≠𝑗) is sufficiently large compared to 𝐾_0𝑗(forall𝑗∈𝑁), according to the constants 𝐴_0𝑘(forall𝑘∈𝑁), the absolute values of the stability abscissas of 𝐴_𝑖𝑗(forall𝑖,𝑗∈𝑁) and norms of 𝐴_0𝑗. If there is only a stable matrix 𝐀₀in the set 𝐴_0𝑗. If there is a unique stable matrix 𝑗∈𝑁, for some ∑_{tk,𝑡𝑘+1∈𝑆𝑇𝜎}(𝑡_𝑘+1−𝑡_𝑘∶𝜎(𝑡_𝑘)=j), then the switched system is globally exponentially stable only if the switching function is such that 𝐴_0𝑗 has infinite measure. If there is a unique stable matrix 𝑗∈𝑁for some 𝑆𝑇_𝜎 and if the sequence of switching instants 𝜎(𝑡_𝑘)=𝑗 is finite, then the switched system is globally exponentially stable if the switching function is such that 𝑡_𝑘 for the last switching instant 𝐋^𝑛2.

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 𝐋^𝑝2 and Γ, respectively. As a result, the input-state Γ₀ and input-output 𝐋(𝐋^𝑚2,𝐋^𝑛2) operators are members of 𝐋(𝐋^𝑚2,𝐋^𝑝2) and 𝜎∶𝐑₀₊→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_ℎ(𝛼))∈𝐋^𝑛2[𝛼,∞) then the unforced state and output trajectory solutions (𝐶Φ𝑥_ℎ(𝛼))∈𝐋^𝑝2[𝛼,∞) and 𝛼∈𝐑₀₊, for all Φ∈𝐋(𝐑^𝑛×[−ℎ,0],𝐋ⁿ₂[𝛼,∞)), respectively. Thus, (𝐶Φ)∈𝐋(𝐑ⁿ×[−ℎ,0],𝐋^p₂[𝛼,∞)) and Φ∈𝐋(𝐑ⁿ×[−ℎ,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×[−ℎ,0],𝐋^p₂) and 𝑢∈𝐋^𝑚2[𝛼,∞).
If, in addition, 𝛼∈𝐑 for some (Γ𝑢_𝛼)∈𝐋^𝑛2[𝛼,∞) then the respective forced solutions fulfil (Γ_𝑜𝑢)∈𝐋^𝑝2[𝛼,∞) and 𝛼∈𝐑₀₊, for all Γ∈𝐋(𝐋^𝑚2[𝛼,∞),𝐋^𝑛2[𝛼,∞)), which are then bounded operators. Thus, Γ_o∈𝐋(𝐋^𝑚2[𝛼,∞),𝐋^𝑝2[𝛼,∞)) and 𝑢∈𝐋^𝑚2+.
If (Γ₊𝑢)∈𝐋^𝑛2+ then the respective forced solutions fulfil (Γ_𝑜+𝑢)∈𝐋^𝑝2+ and Γ₊∈𝐋(𝐋^𝑚2+,𝐋^𝑛2+), respectively, so that Γ_o+∈𝐋(𝐋^𝑚2+,𝐋^𝑝2+) and 𝑢∈𝐋^𝑚2∩𝐵𝑃𝐶⁽⁰⁾(𝐑₀₊,𝐑^𝑚). Equivalently, if 𝑢(𝑡)=0, that is, 𝑡∈𝐑₋, for all Γ₊∈𝐋(𝐋^𝑚2+,𝐋^𝑛2+), then Γ_o+∈𝐋(𝐋^𝑚2+,𝐋^p₂₊) and 𝑢∈𝐋^𝑚2∩𝐵𝑃𝐶⁽⁰⁾(𝐑₀₊,𝐑^𝑚). Equivalently, if Γ∈𝐋(𝐋^𝑚2,𝐋^𝑛2), then Γ_o+∈𝐋(𝐋^𝑚2,𝐋^𝑝2) and 𝑢∈𝐋^𝑚2−.
If Theorem 2.2 holds and (Γ₋𝑢)∈𝐋^𝑛2−, then the respective forced solutions fulfil (Γ_o−𝑢)∈𝐋^𝑝2− and Γ₋∈𝐋(𝐋^𝑚2−,𝐋^𝑛2−)so that Γ_o−∈𝐋(𝐋^𝑚2−,𝐋^𝑝2−)and 𝑢∈𝐋^𝑚2∩𝐵𝑃𝐶⁽⁰⁾(𝐑₀₋,𝐑^𝑚). Equivalently, if Γ∈𝐋(𝐋^𝑚2,𝐋^𝑛2), then Γ_o+∈𝐋(𝐋^𝑚2,𝐋^𝑝2) 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 𝐋^𝑚2 on its appropriate definition domains.
If the system is globally asymptotically stable, then it is possible to restrict the domain and image 𝐋^𝑛2 and Γ₊ of 𝐋^𝑚2+ to 𝐋^𝑛2+ and 𝑢∈𝐋^𝑚2+, respectively, for vector functions (Γ₊𝑢)∈𝐋ⁿ₂₊ such that 𝐑₀₊, since their support is (Γ𝑢)(𝑡)=0,forall𝑡∈𝐑₀₋ and Γ₊∈𝐋(𝐋^m₂₊,𝐋ⁿ₂₊), and then to define a restricted operator (𝐶Γ+𝐷)₊∈𝐋(𝐋^𝑚2+,𝐋^𝑝2+). In the same way, it is possible to define a restricted operator Γ₋∈𝐋(𝐋^𝑚2−,𝐋^𝑛2−). Similarly, it is possible to define (𝐶Γ+𝐷)₋∈𝐋(𝐋^𝑚2−,𝐋^𝑝2−) and 𝑢∈𝐋^𝑚2− of usefulness for vector functions 𝐑_𝑦≡𝐑_± 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]∶𝐑^𝑚×[0,t]→𝐑^𝑛×[0,𝑡]

This section investigates the input-state and input-output operators Γ_𝑜[0,𝑡]∶𝐑^𝑚×[0,t]→𝐑^𝑝×[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 Γ∈𝐋(𝐿^𝑚2,𝐿^𝑛2), further conditions for Γ_𝑜∈𝐋(𝐿^𝑚2,𝐿^𝑝2) and Γ_𝐑𝑦∈𝐋(𝐿^𝑚2[𝐑_𝑦],𝐿^𝑛2[𝐑_𝑦]) are investigated and weaker ones are also given for Γ_𝑜𝑅𝑦∈𝐋(𝐿^𝑚2[𝐑_𝑦],𝐿^𝑝2[𝐑_𝑦]) or 𝐑_𝑦⊂𝐑 with 𝐑_𝑦≡𝐑_± being a bounded real interval, in particular for 𝐑_𝑦≡𝐑_±. Finally, The Hankel and Toeplitz causal and anticausal operators are investigated concerning the cases 𝑢∈𝐋^𝑚2∩𝐵𝑃𝐶⁽⁰⁾(𝐑,𝐑^𝑚). 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. 𝐴_0𝑗, and the matrices 𝐴^−0𝑗 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 𝑗∈𝑁, for all  min_𝑗∈𝑁(|𝜌_0𝑗|,|𝜌̃_0𝑗|)≥𝜀∈𝐑₊, 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 𝐴_0𝑗 are stable (resp., antistable).

Assumption 3.2. Assumption 3.1 holds and, furthermore, the matrices 𝑗∈𝑁 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 𝑇_𝑗 of the complex plane from Theorems 2.1 and 2.2. Furthermore, there exist nonunique coordinate transformations 𝑗∈𝑁, for all 𝐴_0𝑗=𝑇_𝑗𝐴_0𝑗𝑇^𝑗−1=BlockDiag𝐴^−0𝑗,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⁺=𝑛−𝑛⁻) and of order 𝑗∈𝑁, for all 𝑇_𝑗=𝑇. Note also that if Assumption 3.2 holds, then 𝑗∈𝑁, for all 𝑥̃(𝑡)=𝑇(𝑡_𝑘)𝑥(𝑡). After a linear change of variables 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), for all 𝑡_𝑘,𝑡_𝑘+1∈𝑆𝑇_𝜎 with 𝜎(𝑡_𝑘)=𝑗, such that 𝑇(𝑡_𝑘)=𝑇_𝑗 and 𝑗∈𝑁, for some 𝑥̇̃(𝑡)=^𝑞_𝑖=0𝐴_𝑖(𝑡)𝑥̃𝑡−ℎ_𝑖+𝐵(𝑡)𝑢(𝑡),𝑦(𝑡)=𝐶(t)𝑥̃(t)+𝐷(𝑡)𝑢(𝑡),(3.2), the system (2.1) may be described as follows: for all 𝐴𝐶_𝑖((𝑡𝑡))==𝐴𝐶_𝑖𝑘𝑘∈𝐀_𝑖∶=𝐴_𝑖𝑗∈𝐑^𝑛×𝑛∶𝑗∈𝑁,𝐵(𝑡)=𝐵_𝑘∈𝐁∶=𝐵_𝑗∈𝐑^𝑛×𝑚∶𝑗∈𝑁,∈𝐂∶=𝐶_𝑗∈𝐑^𝑝×𝑛∶𝑗∈𝑁,𝐷(𝑡)=𝐷_𝑘∈𝐃∶=𝐷_𝑗∈𝐑^𝑝×𝑚∶𝑗∈𝑁,(3.3), where

for some 𝜎(𝑡)=𝑘 such that 𝐴_𝑖(𝑡)=𝐴_𝑖𝑗=𝑇_𝑗𝐴_𝑖𝑗𝑇^𝑗−1=𝐴^{−−𝑖𝑗}𝐴^−+𝑖𝑗𝐴^+−𝑖𝑗𝐴^++𝑖𝑗,∀𝑖∈𝐵(𝑡)=𝐵_𝑗𝑞∪{0},andsome𝑗∈𝑁,(3.4)=𝑇_𝑗𝐵_𝑗=𝐵^−𝑗𝐵^+𝑗=𝐶(𝑡)=𝐶^𝑗=𝑇_𝑗𝐶_𝑗𝑇^𝑗−1=BlockDiag𝐶^−𝑗,𝐶^+𝑗,𝐷(𝑡)=𝐷^𝑗=𝐷_𝑗,(3.5) subject to (3.1) and (3.4) below: for all 𝐴^−0𝑗𝐴∈_++𝑖𝑗𝐑^{𝑛−×𝑛−},𝐴^−0𝑗∈𝐑^𝑛+×𝑛+,𝐴^{−−𝑖𝑗}∈𝐑^{𝑛−×𝑛−},𝐴^−+𝑖𝑗∈𝐑^{𝑛−×𝑛+},𝐴^+−𝑖𝑗∈𝐑^{𝑛+×𝑛−},∈𝐑^𝑛+×𝑛+,𝐵^−𝑗∈𝐑∀^{𝑛−×𝑚}𝑖,∈𝐵^+𝑗∈𝐑^𝑛+×𝑚,𝐶^−𝑗∈𝐑^{𝑝×𝑛−},𝐶^+𝑗∈𝐑^𝑝×𝑛+,𝑞∪{0},∀𝑗∈𝑁.(3.6) and some for all 𝜒⁻(𝐴_0𝑗)=Im𝑇^𝑗−1[^𝐼𝑛−₀] where

The subspaces 𝑇_𝑗 and 𝐴_0𝑗 are independent of 𝑗∈𝑁 and are called, respectively, the stable and antistable subspaces of 𝐑^𝑛=𝜒⁻(𝐴_0𝑗)⊕𝜒⁺(𝐴_0𝑗), for all 𝑗∈𝑁, which are complementary, that is, 𝐑^𝑛=𝜒⁻(𝐴₀(𝑡))⊕𝜒⁺(𝐴₀(𝑡)), for all 𝑡∈𝐑⁺⁰, so that Π^−𝑗=𝑇^𝑗−1𝐼^𝑛−000𝑇_𝑗,Π^+𝑗=𝑇^𝑗−100𝐼0_𝑛+𝑇^𝑗Π,∀𝑗∈𝑁,₋(t)=𝑇⁻¹(𝑡)𝐼^𝑛−000𝑇(𝑡)=𝑇_𝑗−1𝐼^𝑛−000𝑇_𝑗Π₊,(𝑡)=𝑇⁻¹(𝑡)𝐼^𝑛−000𝑇(𝑡)=𝑇_𝑗−1𝐼^𝑛−000𝑇_𝑗,(3.7), for all Π^±𝑗_𝑘=Π^±(𝑡)=Π^±(𝑡_𝑘). The projections on those subspaces are given by the respective formulas:

and 𝜎(𝑡)=𝑗_𝑘 for somet∈[𝑡_𝑘,𝑡_𝑘+1) such that 𝑡_𝑘,𝑡_𝑘+1∈ST(𝜎); for all 𝛼=0 for each ẋ̃(t)=𝑇(𝑡)𝑥(𝑡)=𝑍(𝑡,0)𝑇(0)𝑥(0)+^𝑞_𝑖=0^0−ℎ_𝑖𝑍(𝑡,𝜏)𝑇(𝜏)𝑥(𝜏)𝑑𝜏+^𝑡−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏,(3.8). Thus, from (2.13)–(2.15), and 𝑥̃=𝜑=𝑇(0)𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛), one gets directly

with initial conditions 𝑍(𝑡,0)∈𝐶⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛), so that 𝐑₀₊, SI_𝜎 is an everywhere differentiable matrix function on 𝑍̇(𝑡)=T𝑡_𝑘𝑍̇(𝑡)=^𝑞_𝑖=0𝐴_𝑖𝑡_𝑘𝑇𝑡−ℎ_𝑖𝑍𝑡−ℎ_𝑖,0,∀𝑡∈𝑡_𝑘,𝑡_𝑘+1,∀𝑡_𝑘∈ST(𝜎),(3.9), with almost everywhere continuous time-derivative except at time instants in 𝐑₊, which satisfies:

on 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), since 𝑍(t,0)=0, for all 𝑡∈𝐑₋, whose unique solution satisfies 𝑍(𝑡,0)=𝑇(𝑡)𝑍(𝑡,0)=Φ(𝑡,0)𝐼_𝑛+^𝑞_𝑖=1^𝑡0Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑇𝜏−ℎ_𝑖𝑍𝜏−ℎ_i,0𝑑𝜏,∀𝑡∈𝐑₀₊.(3.10), for all 𝑒𝐴^0(𝑡)𝑡=𝑇(𝑡)𝑒^{𝐴0(𝑡)𝑡}𝑇⁻¹(𝑡)=BlockDiag𝑒𝐴^{−0(𝑡)𝑡},𝑒𝐴^+0(𝑡)𝑡,(3.11a), and is defined by

Then, so that

for all 𝑡_𝑘,𝑡_𝑘+1∈ST_𝜎, with 𝑒𝐴^0𝑗𝑡=𝑇_𝑗𝑒^{𝐴0𝑗𝑡}𝑇^𝑗−1=BlockDiag𝑒𝐴^−0𝑗𝑡,𝑒𝐴^+0𝑗𝑡Φ𝑡,𝑡_𝑘,=𝑇_𝑗Φ𝑡,𝑡_𝑘𝑇^𝑗−1;𝑍𝑡,𝑡_𝑘=𝑇_𝑗𝑍𝑡,𝑡_𝑘𝑇^𝑗−1,(3.12), for all 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), for all 𝑡_𝑘∈ST(𝜎) since

for all 𝜎∶𝐑₀₊→𝑁, for all 𝑇_𝜎(t) provided that (𝚪𝑢)(𝑡)==^𝑡−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_𝑡−∞Φ(𝑡,𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞=Φ(0,𝛾)𝐴_𝑖(𝛾)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏()3𝑑.1𝜏3a)_𝑡−∞=𝑍(𝑡,𝜏)Π⁻(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏−^∞𝑡𝑍(𝑡,𝜏)Π⁺(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏(3.13b)_𝑡−∞Φ(𝑡,𝜏)Π⁻(𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞×𝑈(𝜏Φ)−(0𝑈,𝛾)𝛾𝐴−_𝑖(𝛾ℎ)_𝑖𝑍𝛾−ℎ_𝑖,𝜏Π⁻(𝜏)𝐵(𝜏)−𝑑𝛾𝑢(𝜏)𝑑𝜏_∞𝑡Φ(𝑡,𝜏)Π⁺(𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞×𝑈(𝜏Φ)−(0𝑈,𝛾)𝛾𝐴−_𝑖(𝛾ℎ)_i𝑍𝛾−ℎ_𝑖,𝜏Π⁺(𝜏)𝐵(𝜏)𝑑𝛾𝑢(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 𝚪_𝑜𝑢(𝑡)==^𝑡−∞𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)_𝑡−∞𝐶(𝑡)Φ(𝑡,𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞×𝑈(𝜏)Φ−(𝑈0,𝛾𝛾)𝐴−_𝑖ℎ(𝛾_𝑖)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)=𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)(3.14a)_𝑡−∞=𝐶(𝑡)𝑍(𝑡,𝜏)Π⁻(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏−^∞𝑡𝐶(𝑡)𝑍(𝑡,𝜏)Π⁺(𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)(3.14b)_𝑡−∞𝐶(𝑡)Φ(𝑡,𝜏)Π⁻(𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞×𝑈(𝜏)−Φ𝑈(0,𝛾𝛾)−𝐴ℎ_𝑖(_𝑖𝛾)𝑍𝛾−ℎ_𝑖,𝜏Π⁻(𝜏)𝐵_𝜎(𝜏)−𝑑𝛾𝑢(𝜏)𝑑𝜏_∞𝑡𝐶(𝑡)Φ(𝑡,𝜏)Π⁺(𝜏)𝐵(𝜏)+^𝑞_𝑖=1^𝑡−∞×𝑈(𝜏)−Φ𝑈(0,𝛾𝛾)−𝐴ℎ_𝑖(_𝑖𝛾)𝑍𝛾−ℎ_𝑖,𝜏Π⁺(𝜏)𝐵(𝜏)𝑑𝛾𝑢(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 ^𝑡−∞Φ(t,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+^𝑞_𝑖=1^{𝜏−ℎ𝑖}_−∞^𝑡0=Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝐵(𝛾)𝑢(𝛾)𝑑𝛾𝑑𝜏_𝑡−∞+Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_𝑞_𝑖=1^𝑡−∞^𝑡0=Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝐵(𝛾)𝑈(𝛾)−𝑈𝜏−ℎ_𝑖𝑢(𝛾)𝑑𝜏𝑑𝛾_𝑡−∞+Φ(𝑡,𝜏)𝐵_𝜎(𝜏)(𝜏)𝑢(𝜏)𝑑𝜏_{𝑡𝑞−∞}_𝑖=1^𝑡0=Φ(𝑡,𝜏)𝐴_𝑖(𝜏)𝑍𝜏−ℎ_𝑖,𝛾𝐵(𝜏)𝑈(𝛾)−𝑈𝜏−ℎ_𝑖𝑢(𝛾)𝑑𝜏𝑑𝛾_𝑡−∞+Φ(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_{𝑡𝑞−∞}_𝑖=1^𝑡0Φ(𝑡,𝛾)𝐴_𝑖(𝜏)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑢(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∈[𝑡_𝑗,𝑡_𝑗+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(𝑡_𝑗) 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 𝑗∈𝑁₀⊂𝐍 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)=𝑇_{𝜎𝜏,𝑡}, 𝜎_𝜏,𝑡∶[𝜏,𝑡)→𝑁_𝜏,𝑡⊂𝑁, Φ(𝑡,𝜏)=BlockDiagΦ⁻(𝑡,𝜏),Φ⁺(𝑡,𝜏)=𝑇(𝑡,𝜏)Φ(𝑡,𝜏)𝑇⁻¹(𝑡,𝜏),(3.19) for 𝜎_𝜏,𝑡(∶=𝜎∣[𝜏,𝑡))∶[𝜏,𝑡)(⊂𝐑₀₊)→N_𝜏,𝑡⊂𝑁.

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 Φ𝜏,𝑡_𝑖==BBlloocckkDDiiaaggΦ𝑒A⁻𝜏,𝑡_𝑖,_{−0𝑡𝑖𝜏−𝑡𝑖}Φ⁺𝜏,𝑡_𝑖=𝑇𝑡_𝑖Φ𝜏,𝑡_𝑖𝑇⁻¹𝑡_𝑖,𝑒A^{+0𝑡𝑖𝜏−𝑡𝑖},∀𝜏∈𝑡^𝑖,𝑡_𝑖+1,∀𝑡_𝑖,𝑡_𝑖+1∈𝑆𝑇_𝜎,(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 𝑇_𝜎(𝑡)=𝑇. 𝑡∈𝐑₀₊ and 𝐀₀ are, in general, of time interval-dependent sizes 𝜎_𝑡≡𝜎_0,𝑡(∶=𝜎∣[0,𝑡))∶[0,𝑡)→𝑁_𝑡⊂𝑁, 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 Φ(𝑡,𝜏)==BBlloocckkDDiiaagg𝑒Φ⁻(_{∑𝑘(𝑡)𝑖=1}𝑡,𝜏),Φ⁺(𝑡,𝜏)=𝑇Φ(𝑡,𝜏)𝑇⁻¹(3.21)A⁻⁰_{𝑡𝑖−1𝑡𝑖−𝑡𝑖−1},𝑒^{∑𝑘(𝑡)𝑖=1}A⁺⁰_{𝑡𝑖−1𝑡𝑖−𝑡𝑖−1},(3.22), 𝑡,𝜏∈𝐑₀₊.
(iii) If Assumption 3.2 holds, then (ii) holds with constant 𝑡_𝑖,𝑡_𝑖+1∈𝑆𝑇_𝜎𝑡, for all Φ^±(𝑡,𝜏)Φ(0,𝛾)𝐴_𝑖(𝛾)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)^±=Φ^±(𝑡,𝛾)𝐴_𝑖(𝛾)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)^±,(3.23).
(iv) If Assumption 3.2 holds, and all the matrices in the set 𝐀_𝑖(forall𝑖∈𝑞) 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 Φ(𝑡,𝜏)𝐴^𝑖(𝜏)𝑍𝛾−ℎ_𝑖,𝜏𝐵(𝜏)^±=Φ^±(𝑡,𝜏)𝐴^±𝑖(𝜏)𝑍^±𝛾−ℎ_𝑖,𝜏𝐵^±(𝜏);∀𝑡,𝜏∈𝐑,(3.24) defining the switched system by the partial switching function up to time Φ(𝜏,𝑡_𝑖) have a block diagonal structure with two block matrices of common sizes 𝐴_{0𝜎(𝑡𝑖)} and 𝐴₀(𝑡_𝑖), then A⁻⁰(𝑡_𝑖)(𝜏−𝑡_𝑖)(A⁺⁰(𝑡_𝑖)(𝜏−𝑡_𝑖)) is block diagonalizable with two nonzero square matrix blocks of time invariant sizes Φ⁻(𝜏,𝑡_𝑖)(Φ⁺(𝜏,𝑡_𝑖)) 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 𝑒^{𝐴10𝑗𝛼}⋅𝑒^{𝐴20𝑗𝛽}=𝑒^{(𝐴0j1𝛼+𝐴20𝑗𝛽)} is an exponential matrix function of 𝑗_1,2∈𝑁_𝑡 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 𝐀_𝑖, 𝑖∈𝑞 and any 𝐀₀ which commute, 𝐏^𝑟±∶𝐋^𝑟2→𝐋^𝑟2±, for all 𝐋^𝑟2. Its second part follows from the semigroup property of 𝐋^𝑟2±.
(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₊∶𝐋^𝑚2+→𝐋^𝑛2+ and Γ₀₊∶𝐋^𝑚2+→𝐋^𝑝2+, under the given assumptions since the matrices Γ₊∶𝐋^𝑚2−→𝐋^𝑛2−, for all Γ₀₊∶𝐋^𝑚2−→𝐋^𝑝2−, are diagonalizable with identical two square matrix blocks of identical sizes.

If all the matrices in the set 𝑡∈𝐑₋ 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 𝑡∈𝐑₋ the usual orthogonal projections of 𝐑₀₊ontoΓ∈𝐋(𝐋^𝑚2,𝐋^𝑛2). 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. Γ_𝑜∈𝐋(𝐋^𝑚2,𝐋^𝑝2) and Γ₊∶𝐋^𝑚2+→𝐋^𝑛2+ are linear bounded, equivalently continuous, operators if any of the properties Theorem 2.1(i)–(iii) holds, and Γ_𝑜+∶𝐋^𝑚2+→𝐋^𝑝2+ and 𝐋^𝑚2 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 𝐋^𝑚2+) and, furthermore, square-integrable, then the state and output trajectory solutions are identically zero for 𝐋^𝑛2 and square-integrable on 𝐋^𝑝2. 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 𝐋^𝑛2+ and 𝐋^𝑝2+, the input-state operators Γ∈𝐋(𝐋^𝑚2,𝐋^𝑛2) and input-output Γ_𝑜∈𝐋(𝐋^𝑚2,𝐋^𝑝2) (identified with the so-called causal Toeplitz operator if the input is an original vector function) have domains restricted from Γ₋∶𝐋^𝑚2−→𝐋^𝑛2− toΓ_𝑜−∶𝐋^𝑚2−→𝐋^𝑝2− and projected images from 𝐋^𝑚2, respectively, 𝐋^𝑚2−, onto 𝐋^𝑛2, respectively,𝐋^𝑝2, provided that 𝐋^𝑛2+ and 𝐋^𝑝2+. In the same way, the input-state operators Γ∈𝐋(𝐋^𝑚2,𝐋^𝑛2) and input-output Γ_𝑜∈𝐋(𝐋^𝑚2,𝐋^𝑝2) have domains restricted from Γ∈𝐋(𝐋^𝑚2,𝐋^𝑛2) to Γ_𝑜∈𝐋(𝐋^𝑚2,𝐋^𝑝2) and projected images from 𝐋^{𝑚,𝑛𝑝2±}, respectively, Γ∈𝐋(𝐋^𝑚2,𝐋^𝑛2), onto Γ_𝑜∈𝐋(𝐋^𝑚2,𝐋^𝑝2), respectively, 𝐇_Γ𝑜∶𝐏^𝑝+Γ_𝑜|_𝐋^𝑚2−=𝐏^𝑝+Γ_𝑜𝐏^𝑚−, provided that Γ_𝑜 and 𝐇_Γ𝑜∶𝐏^𝑝−Γ_𝑜|_𝐋^𝑚2+=𝐏^𝑝−Γ_𝑜𝐏^𝑚+. 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 Γ_𝑜 and 𝐓_Γ𝑜∶𝐏^𝑝+Γ_𝑜|_𝐋^𝑚2+=𝐏^𝑝+Γ_𝑜𝐏^𝑚+ subject to domain restrictions and orthogonal projections of their images since they act on half axis Lebesgue spaces Γ_𝑜.

Definition 3.7. Let 𝐓_Γ𝑜∶𝐏^𝑝−Γ_𝑜|_𝐋^𝑚2−=𝐏^𝑝−Γ_𝑜𝐏^𝑚− be bounded, so that Γ_𝑜 is also bounded. We define the following:(1)the causal input-output Hankel operator (or, simply causal Hankel operator) 𝐇_Γ∶𝐏^𝑛+Γ|_𝐋^𝑚2−=𝐏^𝑛+Γ𝐏^𝑚− with symbol Γ,(2)the anticausal input-output Hankel operator (or, simply anticausal Hankel operator) 𝐇_Γ∶𝐏^𝑛−Γ|_𝐋^𝑚2+=𝐏^𝑛−Γ𝐏^𝑚+ with symbol Γ,(3)the causal input-output Toeplitz operator (or, simply causal Toeplitz operator) 𝐓_Γ∶𝐏^𝑛+Γ|_𝐋^𝑚2+=𝐏^𝑛+Γ𝐏^𝑚+ with symbol Γ,(4)the anticausal input-output Toeplitz operator (or, simply anticausal Toeplitz operator) 𝐓_Γ∶𝐏^𝑛−Γ|_𝐋^𝑚2−=𝐏^𝑛−Γ𝐏^𝑚− with symbol Γ,(5)the causal input-state Hankel operator 𝐓_Γ𝑜+𝐇_Γ𝑜=Γ_𝑜𝐏^𝑚+ with symbol 𝐓_Γ𝑜=𝐏^𝑝+Γ_𝑜𝐏^𝑚+,(6)the anticausal input-state Hankel operator 𝐇_Γ0=𝐏^𝑝−Γ_𝑜𝐏^𝑚+=0 with symbol 𝚪_𝑜𝐏^𝑚+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 𝐓_Γ𝑜+𝐇_Γ𝑜=Γ_𝑜𝐏^𝑚− with symbol 𝐓_Γ𝑜=𝐏^𝑝−Γ_𝑜𝐏^𝑚−,(8)the anticausal input-state Toeplitz operator 𝐇_Γ𝑜=𝐏^𝑝+Γ_𝑜𝐏^𝑚−=0 with symbol 𝚪_𝑜𝐏^𝑚−𝑢(𝑡)=^0−∞𝐶(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 [1–4]), 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) 𝐓_Γ+𝐇_Γ=Γ𝐏^𝑚+, so that 𝐓_Γ=𝐏^𝑝+Γ𝐏^𝑚+ if and only if 𝐇_Γ=𝐏^𝑝−Γ𝐏^𝑚+=0, with 𝐓_Γ+𝐇_Γ=Γ𝐏^𝑚−, so that 𝐓_Γ=𝐏^𝑝−Γ𝐏^𝑚− if and only if 𝐇_Γ=𝐏^𝑝+Γ𝐏^𝑚−=0, with
(ii) (𝐇_Γ0𝑢)(𝑡)=0, so that 𝐓_𝚪𝑜𝑢(𝑡)==𝐏^𝑝+𝚪_𝑜𝐏^𝑚−𝑢(𝑡)_𝑡0=𝐶(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=1^𝑡𝑗_𝑡𝑗−1𝐶𝑡_𝑗−1Φ(𝑡,𝜏)𝐵𝑡_𝑗−1+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1×𝐵𝑡_𝑗−1Φ(𝑡,𝛾)𝐴_𝑖𝑡_ℓ−1𝑍𝛾−ℎ_𝑖,𝜏=𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)_𝑘(𝑡)_𝑗=1^𝑡𝑗_𝑡𝑗−1𝐶𝑡_𝑗−1Φ(𝑡,𝜏)𝐵𝑡_𝑗−1+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1×𝐵𝑡Φ(𝑡,𝛾)𝐴_𝑖𝑡_ℓ−1𝑍𝛾−ℎ_𝑖,𝜏_𝑗−1+𝐷(𝑡)𝑢(𝑡),∀𝑡∈𝐑₀₊𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏.(3.29) if and only if 𝑡₁=0, with 𝑡₁>0, so that 𝑆𝑇_𝜎 if and only if 𝑡₁→0, with
(iii) 𝑖≥1, The last expression being valid if 𝑡∈(−∞,𝑡₂] since 𝐇𝐓_𝚪𝑜𝑢(𝑡)=0,_𝚪𝑜𝑢(𝑡)==−𝐏^𝑝−_∞0𝚪_𝑜𝐏^𝑚+𝑢(𝑡)=−_∞𝐶⁺(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑢(𝜏)𝑑𝜏_𝑗_=𝑘(𝑡)^𝑡𝑗+1_𝑡𝑗𝐶⁺(𝑡)Φ⁺(𝑡,𝜏)𝐵⁺𝑡_𝑗+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1Φ(𝑡,𝛾)𝐴^𝑖𝑡_ℓ−1𝑍𝛾−ℎ_𝑖,𝜏𝐵𝑡_𝑗⁺×𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₋,(3.30). If 𝑡₁=0, then the given switching sequence 𝜎(𝑡)=𝜎(−∞) may be redefined as 𝑡∈(−∞,𝑡₂], (𝐓_Γ𝑜𝑢)(𝑡)=0 for all 𝐇_𝚪𝑜𝑢(𝑡)==𝐏^𝑝+𝚪_𝑜𝐏^𝑚−𝑢(𝑡)_0−∞=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑_0−∞=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_0−∞𝐶(−∞)Φ(𝑡,𝜏)𝐵(−∞)+^𝑞_𝑖=1^0−∞×𝐵(−∞)𝑈Φ(𝜏()𝑡,−𝛾)𝑈𝐴_𝑖𝛾(−−∞ℎ)_𝑖𝑍𝛾−ℎ_𝑖,𝜏=𝑑𝛾𝑢(𝜏)𝑑𝜏_0−∞𝐶(−∞)Φ(𝑡,𝜏)𝐵(−∞)+^𝑞_𝑖=1^0−∞×𝐵(−∞)𝑈(𝜏Φ)−(𝑡,𝑈𝛾)𝐴𝛾_𝑖−(−ℎ∞_𝑖)𝑍𝛾−ℎ_𝑖,𝜏𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊,(3.31) with the switching function initialized as (𝐇_Γ𝑜𝑢)(𝑡)=0, for all 𝐓^𝚪_𝑜𝑢(𝑡)==𝐏^𝑝−𝚪_𝑜𝐏^𝑚−𝑢(𝑡)_0−∞=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)_0−∞=𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)_0−∞𝐶⁻(−∞)Φ⁻(𝑡,𝜏)𝐵⁻(−∞)+^𝑞_𝑖=1^0−∞Φ(𝑡,𝛾)𝐴_𝑖(−∞)𝑍𝛾−ℎ_𝑖,𝜏𝐵(−∞)⁻=×𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡)_0−∞𝐶⁻(−∞)Φ⁻(𝑡,𝜏)𝐵⁻(−∞)+^𝑞_𝑖=1^0−∞Φ(𝑡,𝛾)𝐴^𝑖(−∞)𝑍𝛾−ℎ_𝑖,𝜏𝐵(−∞)⁻×𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏+𝐷(𝑡)𝑢(𝑡),∀𝑡∈𝐑₀₋(3.32), so that the switched system is not modified and the above expression is valid for the causal Toeplitz operator: with the switching time instants being redefined with 𝐓_𝚪𝑢(𝑡)==𝐏^𝑛+𝚪𝐏^𝑚−𝑢(𝑡)=^𝑡0𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏=^𝑡0𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_𝑘(𝑡)_𝑗=1^𝑡𝑗_𝑡𝑗−1Φ(𝑡,𝜏)𝐵𝑡^𝑗−1+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1×𝐵𝑡_𝑗−1Φ(𝑡,𝛾)𝐴_𝑖𝑡_ℓ−1𝑍𝛾−ℎ_𝑖,𝜏=𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏_𝑘(𝑡)_𝑗=1Φ(𝑡,𝜏)𝐵𝑡^𝑗−1+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1×𝐵𝑡Φ(𝑡,𝛾)𝐴_𝑖𝑡_ℓ−1𝑍𝛾−ℎ_𝑖,𝜏_𝑗−1𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊.(3.33), so that 𝑡₁=0, for all 𝑡₀=−∞, as above, in the case that the first switching time instant is nonzero:
(iv) 𝑡₁>0, 𝑡₁→0,𝑡_𝑖+1→𝑡_𝑖,…,
(v) 𝜎(𝑡)=𝜎(𝑡₀)=−∞ The last expression being valid if 𝐇𝐓_𝚪𝑢(𝑡)=0,_𝚪𝑢(𝑡)==−𝐏^𝑛−_∞𝚪𝐏^𝑚+𝑢(𝑡)=−^∞0𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑢(𝜏)𝑑𝜏_𝑗_=𝑘(𝑡)^𝑡𝑗+1_𝑡𝑗Φ⁺(𝑡,𝜏)𝐵⁺𝑡_𝑗+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1Φ(𝑡,𝛾)𝐴^𝑖𝑡_ℓ−1𝑍𝛾−ℎ_𝑖,𝜏𝐵𝑡_𝑗⁺×𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₋,(3.34) since 𝑡₁=0. If 𝜎(𝑡)=𝜎(𝑡₀)=𝜎(−∞), then the given switching sequence 𝑡∈(−∞,𝑡₂] may be redefined as (𝐓_Γ𝑢)(𝑡)=0, for all 𝐇_𝚪𝑢(𝑡)==𝐏^𝑛+𝚪𝐏^𝑚−𝑢(𝑡)=^0−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑=^0−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_0−∞Φ(𝑡,𝜏)𝐵(−∞)+^𝑞_𝑖=1^0−∞×𝐵(−Φ∞(𝑡,)𝛾)𝑈𝐴(_𝑖𝜏()−−∞𝑈)𝑍𝛾−𝛾−ℎ_𝑖ℎ_𝑖,𝜏=𝑑𝛾𝑢(𝜏)𝑑𝜏_0−∞Φ(𝑡,𝜏)𝐵(−∞)+^𝑞_𝑖=1^0−∞×𝐵(−∞)Φ𝑈((𝑡𝜏,𝛾))−𝐴𝑈_𝑖(−𝛾∞−)ℎ𝑍_𝑖𝛾−ℎ_𝑖,𝜏𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊,(3.35) with the switching function initialized as (𝐇_Γ𝑢)(𝑡)=0, for all 𝐓^𝚪𝑢(𝑡)=𝐏^𝑛−𝚪𝐏^𝑚−𝑢(𝑡)=^0−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏=^0−∞=𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_0−∞Φ(𝑡,𝜏)𝐵(−∞)+^𝑞_𝑖=1^0−∞×𝐵(−∞Φ)(𝑡,𝑈𝛾)(𝜏𝐴)_𝑖−(−𝑈∞)𝛾𝑍−ℎ𝛾_𝑖−ℎ_𝑖,𝜏=𝑑𝛾𝑢(𝜏)𝑑𝜏_0−∞Φ(𝑡,𝜏)𝐵⁻(−∞)+^𝑞_𝑖=1^0−∞×𝐵(−∞Φ)(𝑈𝑡,𝛾()𝜏𝐴)−_𝑖(−𝑈∞𝛾)𝑍−ℎ𝛾_𝑖−ℎ_𝑖,𝜏𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₋.(3.36), so that the switched system is not modified and the above expression is valid for the causal input-state Toeplitz operator: with the switching time instants being redefined with Γ_𝑜∈𝐋(𝐋^𝑚2,𝐋^𝑝2), so that (−𝜀,𝜀)×𝐑⊂𝐂, for all 𝑇∈𝐑^𝑛×𝑛, as above, in the case that the first switching time instant is nonzero:
(vi) 𝐵(𝑡)=𝐵_{𝜎𝑡𝑘}𝐴(𝑡)=𝐴_{𝜎𝑡𝑘}=𝑇𝐴(𝑡)𝑇⁻¹=𝑇𝐴_{𝜎𝑡𝑘}𝑇⁻¹,=𝑇𝐵(𝑡)=𝑇𝐵_{𝜎𝑡𝑘},𝐶(𝑡)=𝐶_{𝜎𝑡𝑘}=𝐶(𝑡)𝑇⁻¹=𝐶_{𝜎𝑡𝑘}𝑇⁻¹,(3.37)𝑡_ℓ∈ST_𝜎,

Proof. It follows directly from Lemmas 3.3, 3.4, and Definition 3.7 by noting that card(ST_𝜎)<∞ and 𝐀₀ are bounded operators from Assumption 3.1 since all configurations of the switched system have no critically stable eigenvalues and, furthermore, no stable or unstable ones within the open vertical strip 𝐀_𝑖(forall𝑖∈𝑞∪{0}) from Theorems 2.1 and 2.2.

Note that, if Assumption 3.2 holds, then Theorem 3.8 holds with a constant transformation of coordinates 𝑛₋ in (3.4), (3.5), that is,

for all 𝐀₀if 𝑁 and there is no 𝐇^𝚪𝑢(𝑡)=−^∞0=−_∞𝑍⁺(𝑡,𝜏)𝐵(𝜏)⁺𝑢(𝜏)𝑑𝜏_𝑗_=𝑘(𝑡)^𝑡𝑗+1_𝑡𝑗Φ⁺(𝑡,𝜏)𝐵⁺𝑡_𝑗+^𝑞_{𝑖=1𝑘(𝑡)ℓ=1}^𝑡ℓ_𝑡ℓ−1×𝐵Φ⁺(𝑡,𝛾)𝐴^+𝑖𝑡_ℓ−1𝑍⁺𝛾−ℎ_𝑖,𝜏₊𝑡_𝑗𝑈(𝜏)−𝑈𝛾−ℎ_𝑖𝑑𝛾𝑢(𝜏)𝑑𝜏;∀𝑡∈𝐑₀₋,(3.38) so that (𝐇_Γ𝑜𝑢)(𝑡)=𝐶⁺(𝑡)(𝐇_Γ𝑢)(𝑡). Theorem 3.8 can be specified as follows under Assumption 3.2 provided that each matrix of delayed dynamics has two block diagonal expressions of the same orders as those of 𝑡∈𝐑₀₋.

Corollary 3.9. Assume that all the matrices in the set of configurations 𝑡₁=0 are block diagonal with two matrix blocks matrices of orders 𝜎(𝑡)=𝜎(𝑡₀)=𝜎(−∞) and 𝑡∈(−∞,𝑡₂] identical to those of the stable and antistable blocks of the matrices in the set Φ(𝑡,𝜏) consisting of the 𝑍(𝑡,𝜏) of delay-free matrices of dynamics. Thus, Theorem 3.8 has the following particular expressions for the anticausal (input-output) Hankel and input-state Hankel operators provided that Assumption 3.2 hold: so that 𝑛₊, for all Γ_𝑜∶𝐋^𝑚2→𝐋^𝑝2, with the switching time instants being redefined with 𝐇_Γ𝑜=0, so that 𝐇_Γ𝑜=0, for all 𝐇_Γ𝑜=0, as above, in the case that the first switching time instant is nonzero.

Proof. It follows directly from Theorem 3.8 and Lemma 3.5(iii)-(iv) since the matrix functions 𝐇_Γ𝑜=0 and Γ_𝑜∶𝐋^𝑚2→𝐋^𝑝2 maintain a two block diagonal structure with matrices of orders Γ∶𝐋^𝑚2→𝐋^𝑛2 and 𝐇_Γ=0 from (3.11a), (3.11b), and (3.11c).

Definitions of causality and anticausality follow.

Definition 3.10. A bounded input-output linear operator 𝐇_Γ=0 is said to be causal (anticausal) if the anticausal Hankel operator is zero, that is, Γ∶𝐋^𝑚2→𝐋^𝑛2 (if the causal Hankel operator is zero, i.e., 𝜎∶𝐑₀₊→𝑁).

Definition 3.11. The switched system (2.1) is said to be causal (anticausal) if Γ_𝑜∶𝐋^𝑚2→𝐋^𝑝2 (𝐀₀) provided that 𝐇_Γ𝑜∶𝐋^𝑚2→𝐋^𝑝2 is bounded.

Definition 3.12. A bounded input-state linear operator 𝐇_Γ∶𝐋^𝑚2→𝐋^𝑛2 is said to be causal (anticausal) if Γ_𝑜∶𝐋^𝑚2→𝐋^𝑝2 ((𝐇_Γ𝑜𝑢)(𝑡)=−∫^∞0𝐶⁺(𝑡)(𝑍(𝑡,𝜏)𝐵(𝜏))⁺𝑢(𝜏)𝑑𝜏=0).

A direct result from Definitions 3.10–3.12 is the following.

Assertion. If 𝑢∈𝐋^𝑚2+ is bounded and causal (anticausal) then the switched system (2.1) is causal (anticausal) but the converse is not true, in general.

Theorem 3.14. The following properties hold under Assumption 3.1 for a given switching function 𝑡∈𝐑₀₋ provided that it obeys a minimum residence time between consecutive switches which exceeds some appropriate minimum threshold.
(i) 𝐀₀ is bounded independent of the delays, and if all the matrices of delay-free dynamics in the set (𝐇_Γ𝑜𝑢)(𝑡)=0 are stable, then the system (2.1) is globally asymptotically stable and causal independent of the delays.
(ii) If 𝑢∈𝐋^𝑚2+ is zero independent of the delays and the switched system (2.1) is uniformly controllable and uniformly observable independent of the delays then it is globally asymptotically Lyapunov’s stable independent of the delays.
(iii) If 𝑡∈𝐑₀₋ is zero independent of the delays and the switched system (2.1) is uniformly controllable independent of the delays then it is globally asymptotically Lyapunov’s stable independent of the delays.

Proof. (i) 0≤𝑛₊(𝑡)<∞ is bounded from Assumption 3.1, Theorems 2.1 and 2.2 if there is a sufficiently large residence time for the given switching function since there is an eigenvalue-free open vertical strip including the imaginary complex axis for any delays. Thus, all the configurations of the switched system are dichotomic independent of the delays if the switching function is subject to a minimum residence time exceeding an appropriate threshold. From Theorem 3.8 (i) and (iv), the system is causal if the anticausal Hankel operator is zero, namely, 𝑛₊(t)≠0, for all 𝐇^𝚪_𝑜𝑢(𝑡)=^∞0⟹0𝐶=⁺(_∞0𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑢(𝜏)𝑑𝜏=0≥𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑢(𝜏)𝑑𝜏_ℓ_𝑖=1^𝑡𝑖+1_𝑡𝑖𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑍(𝑡,𝜏)𝐵(𝜏)^+𝑇𝑔𝑡_𝑖𝑑𝜏≠0,∀𝑡∈𝐑₀₋,(3.39), for all 𝑢(𝜏)=(𝑍(𝑡,𝜏)𝐵(𝜏))^+𝑇𝑔(𝑡_𝑖). Property (i) follows since if 𝜏∈[𝑡_𝑖,𝑡_𝑖+1) is a set of stable matrices then the switched system is globally asymptotically stable independent of the delays from Theorem 2.2 and Assumption 3.1 and causal from 𝑡_𝑖,𝑡_𝑖+1∈ST_𝜎, for all {0≠𝐑^𝑛϶𝑔(𝑡_𝑖)=𝑜(1/‖(𝑍(𝑡,𝜏)𝐵(𝜏))‖_{𝐋2[𝑡𝑖,𝑡𝑖+1)})}^∞0, for all 𝑡_𝑖+1≥𝑡_𝑖+𝑇. The above factorization exists since 𝑇∈𝐑₀₊ (the number of unstable eigenvalues of any configuration of (2.1) is finite) since the characteristic quasipolynomials of all the configurations have a principal term in view of the structure of (2.1), [17]. Since the system is uniformly observable, then the following contradiction is stated if ∫^{𝑡𝑖+𝑇0}_𝑡𝑖(𝑍(𝑡,𝜏)𝐵(𝜏))(𝑍(𝑡,𝜏)𝐵(𝜏))^+𝑇𝑑𝜏: provided that the control 𝑡_𝑖,𝑡_𝑖+1∈ST_𝜎, for all 𝑇₀∈𝐑₀₊, for all 𝑡_𝑖 for 𝑡_𝑖∈ST_𝜎. The contradiction follows since 𝑛₊(𝑡)=0 for some 𝐇^𝚪𝑢(𝑡)=^∞0𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑢(𝜏)𝑑𝜏=0.(3.40) so that the controllability Grammian𝐀₀, for all 𝜎∶𝐑₀₊→𝑁 is positive definite, for all Γ_𝑜∶𝐋^𝑚2→𝐋^𝑝2 for some constant 𝜎∶𝐑₀₊→𝑁 (independent of 𝐀₀) and for all ℎ_𝑖∈[0,ℎ) if the system (2.1) is uniformly controllable. Thus, 𝑖∈𝑞, and Property (ii) follows. Property (iii) follows in a similar way by neglecting the controllability condition since

The following result strengths Theorem 3.14 since Assumption 3.2 allows to maintain all the eigenvalues strictly outside the imaginary axis independent of the delays via arbitrary switching (see Theorems 2.1 and 2.2).

Corollary 3.15. If Assumption 3.2 holds, then Theorem 3.14 holds for an arbitrary switching function, that is, without requiring a minimum residence time in-between any two consecutive active parameterizations.

Theorem 3.14 has the following simpler version for zero and small delays which follows from the continuity of the eigenvalues with respect to the delays. It is not required that the matrices describing the delayed dynamics of the various configurations have sufficiently small norms compared with the minimum absolute stability abscissa among the configurations associated with the delay-free dynamics defined by the set 𝐇_Γ𝑜∶𝐋^𝑚2→𝐋^𝑝2.

Theorem 3.16. Assume that (a)the set of Assumption 3.1 holds except the stability conditions in Theorem 2.1 (i.e., there is no requirement on the smallness of the norms of the matrices describing the delayed dynamics of the various configurations of the switched system),(b)a switching function ℎ_i∈[0,ℎ) is given which respects a minimum residence time between consecutive switches exceeding some appropriate minimum threshold. Then, the following properties hold.
(i) If 𝑖∈𝑞 is bounded for some switching function ℎ∈𝐑₊ and if all the matrices of delay-free dynamics in the set ℎ_𝑖∈[0,ℎ) are stable, then the system (2.1) is globally asymptotically stable and causal for 𝑖∈𝑞,  for all  𝐇_Γ∶𝐋^𝑚2→𝐋^𝑛2for some sufficiently small ℎ_i∈[0,ℎ).
(ii) If 𝑖∈𝑞 is zero and the switched system (2.1) is uniformly controllable and uniformly observable for ℎ∈𝐑₊, for all ℎ_𝑖∈[0,ℎ) for some sufficiently small 𝑖∈𝑞, then it is globally asymptotically Lyapunov’s stable for Γ_𝑜𝑡∶𝐋^𝑚2𝑡→𝐋^𝑝2𝑡, for all 𝑡∈𝐑.
(iii) If Γ_𝑜∶𝐋^𝑚2→𝐋^𝑝2 is zero and the switched (2.1) is uniformly controllable for 𝐋^𝑟2𝑡∶={𝑓∈𝐋^𝑟2∶𝑓(𝜏)=0,forall𝜏∈(−∞,𝑡)∪(𝑡,∞)}, for all 𝑟 for some sufficiently small [−𝑡,𝑡], then it is globally asymptotically Lyapunov’s stable for Γ_ot, for all 𝑡.

The following result follows from Theorem 3.16 under Assumption 3.2 in the same way as Corollary 3.15 is a consequence of Theorem 3.14.

Corollary 3.17. If Assumption 3.2 holds, then Theorem 3.16 holds for an arbitrary switching function, that is, without requiring a minimum residence time in-between any two consecutive active parameterizations.

The condition of the auxiliary unforced delay-free system being dichotomic can be removed to conclude global asymptotic stability under causality and uniform controllability and observability as proved in the sequel.

Corollary 3.18. If the switched system (2.1) is causal, uniformly controllable, and uniformly observable independent of the delays for a given switching function, then it is globally asymptotically stable independent of the delays.

Proof. Define the truncated linear operator 𝑛 for arbitrary (but finite) 𝑇(t,⋅)∶[𝑡,𝑡+𝑇₀]→𝐑^𝑛×𝑛 in the same way as 𝑍(𝑡,𝜏) where 𝑍(𝑡,𝜏) is the set of square-integrable 𝐇^𝚪_o𝑢(𝑡)=⟹0,𝐇_𝚪≥𝑢(𝑡)=^∞0𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑍(𝑡,𝜏)𝐵(𝜏)^+𝑇𝑢(𝜏)𝑑𝜏=0_{𝑡+𝑇0𝑡}𝑍(𝑡,𝜏)𝐵(𝜏)⁺𝑍(𝑡,𝜏)𝐵(𝜏)^+𝑇𝑔(𝑡)𝑑𝜏=0,(3.41)-real vector functions of compact support 𝑡∈𝐑₀₊. Note that the linear operator 𝑇₀∈𝐑₊ is bounded for any finite time “𝑔∈𝐋_2,𝑡+𝑇0⊃𝐋_2,𝑡+𝛿” irrespective of the spectrum of the system. If the system is causal, uniformly observable, and uniformly controllable, then for some point nonsingular 𝛿∈[0,𝑇₀]-transformation matrix function 0≠𝑢(𝜏)=(𝑍(𝑡,𝜏)𝐵(𝜏))^+𝑇𝑔∈𝐋_2,𝑡+𝑇0 (see Lemma 3.5(i), a matrix function 𝜏∈[𝑡,𝑡+𝑇₀] being similar to 𝑛⁺∶[𝑡,𝑡+𝑇₀]→𝐙₀₊, may be calculated leading to for some any finite 𝑇₀, some constant ∫^𝑡+𝑇0_𝑡(𝑍(𝑡,𝜏)𝐵(𝜏))⁺(𝑍(𝑡,𝜏)𝐵(𝜏))^+𝑇𝑔(𝑡)𝑑𝜏, and some 𝑛⁺(𝑡)∶=max(𝑛⁺(𝜏)∶𝜏∈[𝑡,𝑡+𝑇₀]), for all 𝑛⁺(𝜏), chosen so that 𝑥(𝑡), for all 𝑢(𝑡). The superscript “+” now includes the contribution of the finite number of unstable and critically unstable modes (since the system is not assumed to be dichotomic), and the integrand is a square matrix function of piecewise constant order 𝑦(𝑡), for any finite real ℎ₁∈[0,∞). Such a matrix order function is finite, since the whole number of critically stable and unstable modes is always finite since all the configurations of the switched system have a principal term in its characteristic quasipolynomial. This, together with the finiteness of 𝑥̇(𝑡)=𝐴_0𝜎(𝑡)𝑥(𝑡)+𝐴₁(𝑡)𝑥𝑡−ℎ₁+𝑒₁𝑢(𝑡)=¹_𝑖=0𝑦(𝑡)=𝑒_𝑇1𝐴_𝑖(𝑡)𝑥𝑡−ℎ_𝑖+𝑒₁𝑢(𝑡),𝑥(𝑡),(4.1), implies that the controllability Grammian ℎ₀=0 may be decomposed in a finite sum of matrices of constant order 𝑒₁=(1,0)^𝑇 completed if necessary with zero blocks for the remaining terms in the sum, the number of additive terms being the number of discontinuities in 1th plus one. This leads again to a contradiction as in Theorem 3.14 and the causal system being uniformly controllable and observable cannot possess critically unstable and unstable modes.

4. Example

Consider the second-order switched system of state vector 𝐑² subject to scalar input 𝐴₀(𝑡)=𝐴₀₁∨𝐴₀₂ and output 𝑡∈𝐑₀₊ and one single delay 𝐴₀₁=−010−.35,𝐴₀₂=−20.5−12.6,(4.2):

and 𝑢∈BPC⁽⁰⁾(𝐑,𝐑), where the control matrix is 𝜎∶𝐑₀₊→2={1,2} which is the {𝑡_𝑖} unity vector in 𝐴₀₁, and 𝐴₀₂, for all 𝐴₁(𝑡)=[^{−2.5𝑎(𝑡)0−1.6}], where:

subject to any function of initial conditions 𝜔=[𝑡]=IntegerPart(𝑡), a control ℎ₁, and a switching function 𝜎(𝑡) which defines the sequence of switching instants 𝐴_0𝑖 for the matrix function of delay-free dynamics. In this system the delay-free matrix function of dynamics is of piecewise continuous entries equal to those of 𝐴₁(𝑡) or 𝑖=1,2, and the matrix of delayed dynamics is 𝑡∈𝐑₀₊ with 𝐴₁∶𝐑₀₊→𝐑² with 𝑍₀₊. This system is asymptotically stable independent of the size of the delay 𝜎₀≡𝜎∶𝐑₀₊→2∶={1,2} for any switching function ST_𝜎0∶={𝑡_0𝑖∈𝐑₀₊}, [30, 31]. Note that the system is parameterized by dichotomic stability matrices 𝐴₀₁ and 𝐴₀₂; 𝜎₁∶𝑡∈𝐑₀₊→[𝑡], for all 𝐴₁∶𝐑₀₊→𝐑² since all the three matrices are stable so that they do not have any critical stable eigenvalues. Finally, note that the above system is more general than that proposed in (2.1) since ST_𝜎1∶={𝑡_1𝑖∈𝐙₀₊} is piecewise continuous with discontinuity points at 𝐴_1𝜎1(t)=𝐴₁(𝑡)=−2.52+(1+1𝑡)^[𝑡]/100−1.6.(4.3). In particular, it can be interpreted as within the class (2.1) if

(a)the switching function 𝑥̇(𝑡)=¹_𝑖=0𝐴_{𝑖𝜎𝑖(𝑡)}𝑥𝑡−ℎ_𝑖+𝑒₁𝑢(𝑡),𝑦(𝑡)=𝑒^𝑇1𝑥(𝑡).(4.4) generates some arbitrary real sequence 𝐑₀₊ governing switches in the delay free dynamics between the matrices Φ(𝑡,𝜏)=Φ(𝑡,𝑡_0𝑖)Φ(𝑡_0𝑖,𝜏) and 𝑡_0𝑖∈ST_𝜎0;(b)the switching function 𝑡∈[𝑡_0𝑖,𝑡_0,𝑖+1) generates switches in the matrix function 𝑡∈[0,𝑡_0𝑖] at switching time instants Φ₁𝑡,𝑡_0𝑖≡Φ𝑡,𝑡_0𝑖=e^{−𝑡−𝑡0𝑖}00.25𝑒𝑒^{−3𝑡−𝑡𝑖}−_{−3𝑡−𝑡0𝑖}𝑒^{−𝑡−𝑡𝑖},(4.5) so that

Thus, the dynamic switched system (4.1) is equivalently written with the two above potentially distinct switching functions as

The fundamental matrix of (2.9) on Φ₂𝑡,𝑡_0𝑖≡Φ𝑡,𝑡_0𝑖=𝑒^{−2.5𝑡−𝑡0𝑖}00.25𝑒𝑒^{−2.5𝑡−𝑡𝑖}−𝑒^{−1.6𝑡−𝑡𝑖}_{−1.6𝑡−𝑡0𝑖},(4.6) is defined recursively by products of matrix functions of the form 𝜎₀(𝑡)=𝜎₀(𝑡_𝑖)=2, for all 𝑡∈[𝑡_0𝑖,𝑡_0,𝑖+1), for all Φ(𝑡,𝜏), for all 𝑡_0𝑖→𝜏 where

if 𝚪𝑢_0𝑡(𝑡)==^𝑡0Φ(𝑡,𝜏)𝑒₁𝑢(𝜏)𝑑𝜏_{𝑡𝑡0𝑘(𝑡)}Φ(𝑡,𝜏)𝑢(𝜏)𝑑𝜏+^𝑘(𝑡)_𝑖=1^𝑡0𝑖_{𝑡0,𝑖−1}Φ(t,𝜏)𝑢(𝜏)𝑑𝜏𝑒₁=+^𝑡𝑡_0𝑘(𝑡)𝑒^{−(𝑡−𝜏)}𝛾_𝜎0(𝜏)+𝑒^{−2.5(𝑡−𝜏)}1−𝛾_𝜎0(𝜏),0^𝑇𝑢(𝜏)𝑑𝜏_𝑘(𝑡)_𝑖=1^𝑡0𝑖_{𝑡0,𝑖−1}𝑒^{−(𝑡−𝜏)}𝚪𝛾_𝜎0(𝜏)+𝑒^{−2.5(t−𝜏)}1−𝛾_𝜎0(𝜏),0^𝑇𝑢(𝜏)𝑑𝜏,_𝑜𝑢_0𝑡(𝑡)=+^𝑡0𝑒^𝑇1Φ(𝑡,𝜏)𝑒₁𝑢(𝜏)𝑑𝜏_𝑘(𝑡)_𝑖=1^𝑡0i_{𝑡0,𝑖−1}𝑒^{−(𝑡−𝜏)}𝛾_𝜎0(𝜏)+𝑒^{−2.5(𝑡−𝜏)}1−𝛾_𝜎0(𝜏)𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊,(4.7), for all 𝑢_0𝑡(𝑡)=𝑢(𝑡), and

if 𝑢_0𝑡(𝑡)=0, for all 𝑡∈𝐑₋, and 𝐷=0 is defined from the two above identities with the replacement 𝐶=𝐵=𝑒₁=(1,0)^𝑇if 𝑘(𝑡)=max(𝑡_0𝑖∈ST_𝜎0∶𝐑₀₊϶𝑡_0𝑖≤𝑡).

In the same way, the input-state and input-output operators of (2.10) are defined as follows:

subject to 𝛾_𝜎0∶[𝑡_0𝑖,𝑡_0,𝑖+1)∩𝐑₀₊→{1,0}, for all 𝛾_𝜎0(𝜏)=1, and 𝜎₀(𝑡_𝑖)=1, for all 𝛾_𝜎0(𝜏)=0, since the input-output interconnection gain is zero; that is, 𝜎₀(𝑡_𝑖)=2 and the output and control vectors are 𝜏∈[𝑡_0𝑖,𝑡_0,𝑖+1), where 𝐑₀₊ is the largest switching instant not exceeding the current time 𝑍(𝑡,0)=Φ(𝑡,0)+^𝑡0Φ(𝑡,𝜏)𝐴₁(𝜏)𝑍𝜏−ℎ₁,0𝑑𝜏,∀𝑡∈𝐑₀₊,(4.8), and 𝑍(𝑡,0)=0 is a binary indicator function defined as 𝑡<0 if Φ(𝑡,𝜏)=𝛾_𝜎0(𝜏)Φ₁(𝑡,𝜏)+(1−𝛾_𝜎0(𝜏))Φ₂(𝑡,𝜏) and 𝑡,𝜏(≤𝑡)∈𝐑₀₊ if 𝐴_0𝑖(𝑖=1,2), for all 𝐴₁(𝑡). Thus, the matrix function of (2.15) which defines the solution of the homogeneous system on 𝑢(𝑡) is given by

subject to 𝑦(𝑡), for all 𝐋₂, and ℎ₁∈[0,∞), for all 𝜎∶𝐑₀₊→2={1,2}, (4.5), (4.6), which are obtained from 𝐴_0𝑖,𝐴₁(𝑡), and the definition of 𝑖=1,2 in this example. If the control input 𝐴_0𝑖(𝑖=1,2) is in 𝑇_𝑖=𝐼₂(𝑖=1,2), then the output 𝐴_0𝑖=𝐴_0𝑖(𝑖=1,2) is also in 𝐴₁(𝑡)=𝐴₁(𝑡) independent of the delay 𝑒₁=𝐶=𝐵=𝐶=𝐵 and for any switching function 𝐑₀₊ since the system (4.1), subject to the given matrices 𝐑₀₊, 𝐇𝐓_𝚪0𝑢(𝑡)=0,_𝚪𝑜𝑢(𝑡)=𝐏₊𝚪_𝑜𝐏₋𝑢(𝑡)=𝑒^𝑇1^𝑡0𝑍(t,𝜏)𝑢(𝜏)𝑑𝜏𝑒₁=^𝑘(𝑡)_𝑗=1^𝑡𝑗_𝑡𝑗−1𝑒^𝑇1Φ(𝑡,𝜏)𝑒¹+^𝑘(𝑡)_ℓ=1^𝑡ℓ_{𝑡ℓ0𝑥0𝑒𝑏00−0𝑥0𝑒𝑏001}Φ(𝑡,𝛾)𝐴₁𝑡_{ℓ0𝑥0𝑒𝑏00−0𝑥0𝑒𝑏001}𝑍𝛾−ℎ₁,𝜏𝑒₁×𝑈(𝜏)−𝑈𝛾−ℎ₁𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊.(4.9), is stable independent of the delay and for arbitrary switching. Since 𝑡₁=0 are stability matrices, the transformation matrices 𝑡₀=−∞ then 𝑡₁>0, ST_𝜎, 𝑡₁→0,𝑡_𝑖+1→𝑡_𝑖,… in (3.1)–(3.5). Now, the Definition 3.7 yields to the following particular application of Theorem 3.8 for this example since Assumption 3.1 holds and the input-state and input-output operators are bounded operators. It is found, as expected since all the matrices are stable, that the anticausal input-output and input-state Hankel operators are zero on 𝑖≥1.

(1) The anticausal input-output Hankel and causal input-output Toeplitz operators on 𝜎(𝑡)=𝜎(𝑡₀)=−∞are

The last expression being valid if 𝐑₀₋ since 𝐇^𝚪_𝑜𝑢(𝑡)=𝐏₋𝚪_𝑜𝐏₊𝑢(𝑡)=−^∞0𝑒^𝑇1𝑍(𝑡,𝜏)𝑒¹⁺𝑢(𝜏)𝑑𝜏=𝐓_𝚪𝑜𝑢(𝑡)=0,∀𝑡∈𝐑₀₋,(4.10). If 𝐑₀₊, then the given switching sequence 𝐓𝐇^𝚪_𝑜𝚪𝑜𝑢(𝑡)=0,∀𝑡∈𝐑₀₊,𝑢(𝑡)==𝐏₊𝚪_𝑜𝐏₋𝑢(𝑡)=^0−∞𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_0−∞𝑒^𝑇1Φ(𝑡,𝜏)𝑒¹+^0−∞Φ(𝑡,𝛾)𝐴₁(−∞)𝑍𝛾−ℎ₁,𝜏𝑒₁×𝑈(𝜏)−U𝛾−ℎ₁𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊.(4.11) may be redefined as (4), for all 𝐑₀₋ with the switching function initialized as 𝐓^𝚪_𝑜𝑢(𝑡)==𝐏₋𝚪_𝑜𝟎𝐱𝟎𝐞𝐛𝟎𝟎𝐏₋𝑢(𝑡)=^0−∞𝐶(𝑡)𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏_0−∞𝑒^𝑇1Φ⁻(𝑡,𝜏)𝑒₁+^0−∞Φ⁻(𝑡,𝛾)𝐴₁(−∞)𝑍𝛾−ℎ₁,𝜏𝑒₁⁻𝐇_𝚪𝑜×𝑈(𝜏)−𝑈𝛾−ℎ₁𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₋,𝑢(𝑡)=0,∀𝑡∈𝐑₀₋.(4.12), for all (5) so that the switched system is not modified and the above expression is valid for the causal Toeplitz operator.

(2) The anticausal input-output Hankel and causal input-output Toeplitz operators on 𝐑₀₊are

as above, in the case that the first switching time instant is nonzero.

(3) The anticausal input-output Toeplitz and causal Hankel input-output operators on 𝑡₁=0 are

𝑡₁>0The anticausal input-output Toeplitz and causal Hankel input-output operators on ST_𝜎are

𝐇^{𝚪𝟎𝐱𝟎𝐞𝐛𝟎𝟎}𝑢(𝑡)=𝐏^𝑛−𝚪𝐏₊𝑢(𝑡)=0,𝐓_𝚪𝑢(𝑡)=0,∀𝑡∈𝐑₀₋,(4.14) The anticausal input-state Hankel and causal input-state Toeplitz operators on   𝑡₁=0 are

The last expression being valid if 𝑡∈(−∞,𝑡₂] since 𝐑₀₊. If 𝐇_𝚪𝑢(𝑡)==𝐏^𝑛+𝚪𝐏₋𝑢(𝑡)=^0−∞𝑍(𝑡,𝜏)𝐵(𝜏)𝑢(𝜏)𝑑𝜏,_0−∞Φ(𝑡,𝜏)𝑒¹+^0−∞Φ(𝑡,𝛾)𝐴_𝑖(−∞)𝑍𝛾−ℎ_𝑖,𝜏𝑒₁𝐓_𝚪×𝑈(𝜏)−𝑈𝛾−ℎ₁𝑑𝛾𝑢(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊,𝑢(𝑡)=0,∀𝑡∈𝐑₀₊.(4.15), then the given switching sequence (8) may be redefined as above.

(6) The anticausal input-state Hankel and causal input-state Toeplitz operators on 𝐑₀₋are

with the switching time instants being redefined with Γ₀∶𝐋₂→𝐋₂, so that Γ∶𝐋₂→𝐋^𝑛2, for all 𝑁, as above, in the case that the first switching time instant is nonzero.

(7) The causal input-state Hankel and anticausal input-state Toeplitz operators on are

The causal input-state Hankel and anticausal input-state Toeplitz operators on are

Note that the bounded input-output and input-state linear operators and are causal since the respective anticausal Hankel operators are zero.

Acknowledgments

The author is very grateful to the Spanish Ministry of Education by its partial support of this work through project DPI2006-00714. The author is also grateful to the Basque Government by its support through GIC07143-IT-269-07, SAIOTEK SPED06UN10, and SPE07UN04. Finally, he is also grateful to the reviewers and editor by their valuable comments.