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

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

M. De la Sen

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 [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 + a n d 𝐑 + denote the positive subsets of 𝐙 , respectively, and 𝐂 + denotes the subset of 𝐂 of complex numbers with positive real part.
𝐙 − a n d 𝐑 − 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 ) ( 𝐑 0 + , 𝑋 ) denotes the set of functions of class 𝐶 ( 𝑖 ) . Also, B P C ( 𝑖 ) ( 𝐑 0 + , 𝑋 ) and P C ( 𝑖 ) ( 𝐑 0 + , 𝑋 ) denote the set of functions in 𝐶 ( 𝑖 − 1 ) ( 𝐑 0 + , 𝑋 ) which, furthermore, possess bounded piecewise continuous constant or, respectively, piecewise continuous constant 𝑖 t h 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 𝑛 2 ≡ 𝐋 𝑛 2 ( 𝐑 ) and endowed with the inner product 𝐿 2 -norm ‖ 𝑓 ‖ 𝐋 𝑛 2 ∫ ∶ = ( ∞ − ∞ ‖ 𝑓 ( 𝜏 ) ‖ 2 2 𝑑 𝜏 ) , for all 𝑓 ∈ 𝐋 𝑛 2 , where ‖ ⋅ ‖ 2 is the ℓ 2 -vector (or Euclidean) norm and its corresponding induced matrix norm. 𝐋 𝑛 2 [ 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 𝑛 2 [ 𝛼 , ∞ ) ∫ ∶ = ( ∞ 𝛼 ‖ 𝑓 ( 𝜏 ) ‖ 2 2 𝑑 𝜏 ) , for all 𝑓 ∈ 𝐋 𝑛 2 [ 𝛼 , ∞ ) . 𝐋 𝑛 2 + ∶ = { 𝑓 ∈ 𝐋 𝑛 2 ∶ 𝑓 ( 𝑡 ) = 0 , f o r a l l 𝑡 ∈ 𝐑 − } and 𝐋 𝑛 2 − ∶ = { 𝑓 ∈ 𝐋 𝑛 2 ∶ 𝑓 ( 𝑡 ) = 0 , f o r a l l 𝑡 ∈ 𝐑 + } are closed subspaces of 𝐋 𝑛 𝟐 ∶ = { 𝑓 ∈ 𝐋 𝑛 2 ∶ 𝑓 ( 𝑡 ) = 0 , f o r a l l 𝑡 ∈ 𝐑 − } → 𝐋 𝑛 𝟐 of respective supports 𝐑 0 + and 𝐑 0 − . Then, 𝐋 𝑛 2 = 𝐋 𝑛 2 + ⊕ 𝐋 𝑛 2 − .

𝐼 𝑛 denotes the 𝑛 th identity matrix.

𝜆 m a x ( 𝑀 ) and 𝜆 m i n ( 𝑀 ) stand for the maximum and minimum eigenvalues of a definite square real matrix 𝑀 = ( 𝑚 𝑖 𝑗 ) .

𝜎 ∶ 𝐑 0 + → 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 ) ( ⊂ 𝐑 0 + ) → 𝑁 𝜏 , 𝑡 ⊂ 𝑁 is the partial switching function with its domain restricted to [ 𝜏 , 𝑡 ] . 𝜎 𝑡 is a notational abbreviation of 𝜎 0 , 𝑡 .

The point constant delays are denoted by ℎ 𝑖 ∈ [ 0 , ℎ ] , f o r a l l 𝑖 ∈ 𝑞 ∪ { 0 } and are, in general, incommensurate, and ℎ 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 = ℎ 0 < ℎ 1 < ℎ 2 < ⋯ < 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 𝑝 are 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 ∈ 𝐑 0 + 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): ∑ ̇ 𝑥 ( 𝑡 ) = ( 𝑞 𝑖 = 0 𝐴 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:  ̇ 𝑥 ( 𝑡 ) = 𝐴 0 ( 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 ∶ 𝐑 0 + → 𝑁 , which generates a strictly ordered sequence of switching time instants 𝑆 𝑇 𝜎 ∶ = { 𝑡 𝑖 ∶ 𝑡 𝑖 + 1 ≥ 𝑡 𝑖 + 𝑇 , f o r a l l 𝑖 ∈ 𝑁 0 ( ⊃ { 1 } ) ⊂ 𝐙 + , 𝑡 1 ∈ 𝐑 0 + } , and which might be equivalently rewritten, since 𝐴 𝑖 ( 𝑡 ) = 𝐴 𝑖 𝜎 ( 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 ∶ 𝐑 0 + → 𝑁 , as

where 𝐑 0 + 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 ∈ B P C ( 0 ) ( [ − ℎ , 0 ] , 𝐑 𝑛 ) , that is, 𝜎 ( 𝑡 ) = 𝑗 ∈ 𝑁 , f o r a l l 𝑡 ∈ 𝐑 − ∪ [ 0 , 𝑡 1 ) . It is assumed that 𝑡 1 ∈ S T 𝜎 , 𝜎 ∶ 𝐑 0 + → 𝑁 , being the first switching instant generated by the switching function ( − ∞ , 𝑡 1 ] ; 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 on 𝐑 0 + . The unique mild solution of the state-trajectory solution, which exists on 𝜑 ∈ B P C ( 0 ) ( [ − ℎ , 0 ] , 𝐑 𝑛 ) according to Picard-Lindeloff theorem for any given 𝑢 ∈ B P C ( 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 + 𝑁 𝑡 ( 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 ∶ S I 𝜎 ∋ 𝑡 𝑖 ≤ 𝑡 } ; 𝑡 𝑘 + 𝑖 ∈ S T 𝜎 , 𝑖 ∈ 𝑁 𝑡 ( 𝛼 ) , for all 𝑡 𝑘 + 𝑁 t ( 𝜎 ) + 1 ∈ S T 𝜎 and 𝜎 ( 𝑡 − 𝑘 ) = 𝑗 ( 𝑡 𝑘 ) ≠ 𝜎 ( 𝑡 + 𝑘 ) = 𝑗 ( 𝑡 𝑘 + 1 ) ∈ 𝑁 . Note that 𝑡 𝑘 , 𝑡 𝑘 + 1 ∈ S T 𝜎 , for all 𝑥 ( 𝑡 ) = 𝚽 𝑥 ℎ ( 𝛼 ) ( 𝑡 ) + 𝚪 𝑢 𝛼 ( 𝑡 ) = Φ ( 𝑡 , 𝛼 ) 𝑥 ( 𝛼 ) + 𝑞 𝑖 = 1 𝑡 𝛼 Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑥 𝜏 − ℎ 𝑖 𝑑 𝜏 + 𝑡 𝛼 Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 , ( 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 𝜎 𝜏 , 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 𝚽 𝑥 ℎ ( 𝛼 ) ( 𝑡 ) + 𝐶 𝚪 𝑢 𝛼 + 𝐷 ( 𝑡 ) = 𝐶 ( 𝑡 ) Φ ( t , 𝛼 ) 𝑥 ( 𝛼 ) + 𝑞 𝑖 = 1 𝑡 𝛼 Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑥 𝜏 − ℎ 𝑖 𝑑 𝜏 + 𝑡 𝛼 Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 + D ( 𝑡 ) u ( 𝑡 ) , ( 2 . 7 ) for all the matrices of the individual parameterizations. The output trajectory solution is

for all 𝜑 ∈ B P C ( 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 𝚽 𝑥 ℎ ( 𝛼 ) ( 𝑡 ) ∶ = Φ ( 𝑡 , 𝛼 ) 𝑥 ( 𝛼 ) + 𝑞 𝑖 = 1 𝑡 𝛼 Φ ( 𝑡 , 𝜏 ) 𝑥 𝜏 − ℎ 𝑖 𝑑 𝜏 , ∀ 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 S T 𝜎 with bounded discontinuities on the set [ 𝛼 , 𝑡 ] ⊂ 𝐑 and is defined on the interval Φ ( 𝑡 , 𝛼 ) = 𝑒 𝐴 0 ( 𝑡 𝑘 + 𝑁 𝑡 ( 𝛼 ) + 1 ) ( 𝑡 − 𝑡 𝑘 + 𝑁 𝑡 ( 𝛼 ) + 1 ) ⎛ ⎜ ⎜ ⎝ 𝑘 + 𝑁 𝑡 ( 𝛼 ) 𝑖 = 1 𝑒 𝐴 0 𝑡 𝑘 + 𝑖 𝑡 𝑘 + 𝑖 + 1 − 𝑡 𝑘 + 𝑖 ⎞ ⎟ ⎟ ⎠ 𝑒 𝐴 0 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 𝜎 Γ + 𝐷 𝜎 , respectively, 𝚪 𝑢 𝛼 𝑡 ( 𝑡 ) ∶ = 𝑡 𝛼 Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 = 𝑡 − ∞ Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 𝛼 𝑡 ( 𝚪 𝜏 ) 𝑑 𝜏 𝑜 𝑢 𝛼 𝑡 ( 𝑡 ) ∶ = 𝑡 𝛼 = 𝐶 ( 𝑡 ) Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 + 𝐷 ( 𝑡 ) 𝑢 ( 𝑡 ) 𝑡 − ∞ 𝐶 ( 𝑡 ) Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 𝛼 𝑡 ( 𝜏 ) 𝑑 𝜏 + 𝐷 ( 𝑡 ) 𝑢 ( 𝑡 ) , ∀ 𝑡 ( ≥ 𝛼 ) , 𝛼 ∈ 𝐑 0 + , ( 2 . 1 0 ) , 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 𝛼 𝛼 − ℎ 𝑖 𝑍 ( 𝑡 , 𝜏 ) 𝑥 ( 𝜏 ) 𝑑 𝜏 + 𝑡 − ∞ 𝑍 ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 𝛼 𝑡 ( 𝜏 ) 𝑑 𝜏 , 𝑦 ( 𝑡 ) = 𝐶 ( 𝑡 ) 𝑍 ( 𝑡 , 𝛼 ) 𝑥 ( 𝛼 ) + 𝑞 𝑖 = 1 𝛼 𝛼 − ℎ 𝑖 𝑍 ( 𝑡 , 𝜏 ) 𝑥 ( 𝜏 ) 𝑑 𝜏 + 𝑡 − ∞ 𝑍 ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 𝛼 𝑡 ( 𝜏 ) 𝑑 𝜏 + 𝐷 ( 𝑡 ) 𝑢 ( 𝑡 ) , ( 2 . 1 3 ) . 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 S I 𝜎 , with almost everywhere continuous time-derivative except at time instants in ̇ 𝑍 ( 𝑡 ) = 𝑞 𝑖 = 0 𝐴 ( 𝑡 ) 𝑍 𝑡 − ℎ 𝑖 , 0 ( 2 . 1 4 ) , which satisfies

on 𝑍 ( 𝑡 , 𝛼 ) = 0 whose unique solution satisfies 𝛼 ( < 𝑡 ) , for all 𝑡 ∈ 𝐑 , 𝐼 𝑍 ( 𝑡 , 𝛼 ) = Φ ( 𝑡 , 𝛼 ) 𝑛 + 𝑞 𝑖 = 1 𝑡 𝛼 Φ ( 𝛼 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 , 𝛼 𝑑 𝜏 , ∀ 𝑡 ( ≥ 𝛼 ) , 𝛼 ∈ 𝐑 ( 2 . 1 5 ) , and is defined by

on any time interval 𝛼 = 0 . Now, take 𝑢 ( 𝑡 ) , and consider that the input 𝐑 is defined on 𝐼 𝑥 ( 𝑡 ) = Φ ( t , 0 ) 𝑛 + 𝑞 𝑖 = 1 𝑡 0 Φ ( 0 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 + , 0 𝑑 𝜏 𝑥 ( 0 ) 𝑞 𝑞 𝑖 = 1 𝑗 = 1 t 0 0 − ℎ 𝑗 Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 + , 𝛾 𝜑 ( 𝛾 ) 𝑑 𝛾 𝑑 𝜏 𝑡 − ∞ Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 + 𝑞 𝑖 = 1 𝑡 0 𝜏 − ℎ 𝑖 − ∞ Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 𝐼 , 𝛾 𝐵 ( 𝛾 ) 𝑢 ( 𝛾 ) 𝑑 𝛾 𝑑 𝜏 = Φ ( 𝑡 , 0 ) 𝑛 + 𝑞 𝑖 = 1 𝑡 0 Φ ( 0 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 + , 0 𝑑 𝜏 𝑥 ( 0 ) 𝑞 𝑞 𝑖 = 1 𝑗 = 1 𝑡 0 0 − ℎ 𝑗 Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 𝜑 + , 𝛾 ( 𝛾 ) 𝑑 𝛾 𝑑 𝜏 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 𝑑 𝛾 𝑢 ( 𝜏 ) 𝑑 𝜏 , 𝑦 ( 𝑡 ) = 𝐶 𝜎 ( 𝑡 ) 𝐼 Φ ( 𝑡 , 0 ) 𝑛 + 𝑞 𝑖 = 1 𝑡 0 Φ ( 0 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 + , 0 𝑑 𝜏 𝑥 ( 0 ) 𝑞 𝑞 𝑖 = 1 𝑗 = 1 𝑡 0 0 − ℎ 𝑗 Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 , 𝛾 𝜑 ( 𝛾 ) 𝑑 𝛾 𝑑 𝜏 + 𝑡 − ∞ + Φ ( 𝑡 , 𝜏 ) 𝐵 ( 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 𝑞 𝑖 = 1 𝑡 0 𝜏 − ℎ 𝑖 − ∞ Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 Φ 𝐼 , 𝛾 𝐵 ( 𝛾 ) 𝑢 ( 𝛾 ) 𝑑 𝛾 𝑑 𝜏 + 𝐷 ( 𝑡 ) 𝑢 ( 𝑡 ) = 𝐶 ( 𝑡 ) ( 𝑡 , 0 ) 𝑛 + 𝑞 𝑖 = 1 𝑡 0 Φ ( 0 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 𝑥 + , 0 𝑑 𝜏 ( 0 ) 𝑞 𝑞 𝑖 = 1 𝑗 = 1 𝑡 0 0 − ℎ 𝑗 Φ ( 𝑡 , 𝜏 ) 𝐴 𝑖 ( 𝜏 ) 𝑍 𝜏 − ℎ 𝑖 + , 𝛾 𝜑 ( 𝛾 ) 𝑑 𝛾 𝑑 𝜏 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 𝑑 𝛾 𝑢 ( 𝜏 ) 𝑑 𝜏 + 𝐷 ( 𝑡 ) 𝑢 ( 𝑡 ) , ( 2 . 1 6 ) . Then, the combination of (2.7) with the substitution of (2.13) in the delayed state and output-trajectory solutions yields

where 𝜎 ∶ 𝐑 0 + → 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 ∃ l i m 𝑡 → ∞ 𝑡 𝑖 ∈ 𝑆 𝑇 𝜎 ( 𝑡 ) 𝐾 0 𝑥 0 2 0 0 𝑑 𝑡 0 𝜎 𝑖 𝑒 1 + 𝜌 0 𝜎 𝑡 𝑖 ℎ − 1 𝜌 𝑡 0 𝜎 𝑖 𝑞 𝑖 = 1 ‖ ‖ 𝐴 𝑡 𝑖 𝜎 𝑖 ‖ ‖ 2 × 𝑒 − ( 𝜌 𝑖 ) 0 𝜎 ( 𝑡 − 𝐾 𝑖 ) 0 𝜎 ( 𝑡 ∑ 𝑞 𝑖 = 1 | | 𝐴 𝑖 ) 𝑖 𝜎 ( 𝑡 | | 2 ) 𝑡 𝑖 + 1 − 𝑡 𝑖 = 0 , ( 2 . 1 7 ) is such that where 𝐑 + ϶ 𝜌 0 𝜎 ( 𝑡 𝑖 ) ∈ { 𝜌 0 1 , 𝜌 0 2 , … , 𝜌 0 𝑁 } and 𝜎 ( 𝑡 𝑖 ) = 𝑗 ∈ 𝑁 if ‖ 𝑒 𝐴 0 𝑖 𝑡 ‖ 2 ≤ 𝐾 0 𝑖 𝑒 − 𝜌 0 𝑖 𝑡 , f o r a l l 𝑖 ∈ 𝑁 are real constants such that 𝐀 0 (i.e., all the matrices in the set 𝑆 𝑇 𝜎 ( 𝑡 ) ∶ = { 𝑡 𝑖 ∈ 𝑆 𝑇 𝜎 ∶ 𝑡 𝑖 ≤ 𝑡 } are stable) with 𝑡 𝑠 ( 𝑡 ) + 1 ∶ = 𝑡 and 𝑡 ∉ 𝑆 𝑇 𝜎 if s ( 𝑡 ) ∶ = c a r d 𝑆 𝑇 𝜎 ( 𝑡 ) , where 𝜎 ∶ 𝐑 0 + → 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 0 𝑗 is such that 𝜌 0 𝑗 > 𝐾 0 𝜎 ( 𝑡 𝑖 ) ∑ 𝑞 𝑖 = 1 ‖ 𝐴 𝑖 𝑗 ‖ 2 are all stable matrices satisfying 𝑗 ∈ 𝑁 , for all m a x 𝑡 𝑖 ∈ 𝑆 𝑇 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ) ≥ 𝑇 , and the residence time at each switching instant satisfies | 𝜌 0 𝑗 | with its lower-bound T being sufficiently large according to the respective absolute values 𝐴 0 𝑗 of the stability (or convergence) abscissas of − 𝜌 0 𝑗 < 0 (i.e., 𝐴 0 𝑗 if all the eigenvalues of − 𝜌 0 𝑗 + 𝜀 , 𝜀 → 0 + are distinct and 𝑗 ∈ 𝑁 , otherwise), for all 𝐴 𝑖 𝑗 ( f o r a l l 𝑖 ∈ 𝑞 , 𝑗 ∈ 𝑁 ) and the norms of the matrices 𝜎 ∶ 𝐑 0 + → 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 0 𝑗 is such that at least one 𝜌 0 𝑗 > 𝐾 0 𝜎 ( 𝑡 𝑖 ) ∑ 𝑞 𝑖 = 1 ‖ 𝐴 𝑖 𝑗 ‖ 2 is a stable matrix satisfying m a x 𝑡 𝑖 , 𝑡 𝑖 + 1 ∈ 𝑆 𝑇 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ∶ 𝜎 ( 𝑡 𝑖 ) = 𝑗 ) , and furthermore, ∑ 𝑡 𝑖 , 𝑡 𝑖 + 1 ∈ 𝑆 𝑇 𝜎 m a x ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ∶ 𝜎 ( 𝑡 𝑖 ) ≠ 𝑗 , 𝜎 ( 𝑡 𝑖 + 1 ) ≠ 𝑗 ) is sufficiently large compared to 𝐾 0 𝑗 ( f o r a l l 𝑗 ∈ 𝑁 ) , according to the constants 𝐴 0 𝑘 ( f o r a l l 𝑘 ∈ 𝑁 ) , the absolute values of the stability abscissas of 𝐴 𝑖 𝑗 ( f o r a l l 𝑖 , 𝑗 ∈ 𝑁 ) , and norms of 𝐴 0 𝑗 . If there is only a stable matrix 𝐀 0 in the set 𝐴 0 𝑗 . If there is a unique stable matrix 𝑗 ∈ 𝑁 , for some ∑ 𝑡 𝑘 , 𝑡 𝑘 + 1 ∈ 𝑆 𝑇 𝜎 ( 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 0 𝑗 has infinite measure. If there is a unique stable matrix 𝑗 ∈ 𝑁 for some 𝑆 𝑇 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 𝐀 0 = 𝐀 0 − ∪ ( 𝐀 0 + ∪ 𝐀 0 ± ) .
(iv) If 𝐀 0 − ≠ ∅ where 𝐀 0 + , 𝐀 0 ± and 𝐀 0 are the sets of stable, unstable, and critically stable matrices in the set 𝜎 ∶ 𝐑 0 + → 𝑁 then the switched system is globally exponentially stable independent of the sizes of the delays if the switching function ∑ 𝑡 𝑖 , 𝑡 𝑖 + 1 ∈ 𝑆 𝑇 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ∶ 𝜎 ( 𝑡 𝑖 ) = 𝑗 , 𝐴 0 𝑗 ∈ 𝐀 0 − ) is such that ∑ 𝑡 𝑖 , 𝑡 𝑖 + 1 ∈ 𝑆 𝑇 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ∶ 𝜎 ( 𝑡 𝑖 ) = 𝑗 , 𝜎 ( 𝑡 𝑖 + 1 ) = 𝑘 , 𝐴 0 𝑗 , 𝐴 0 𝑘 ∉ 𝐀 0 − ) is sufficiently large compared to 𝐾 0 𝑗 ( f o r a l l 𝑗 ∈ 𝑁 ) according to the constants 𝐴 0 𝑘 ( f o r a l l 𝑘 ∈ 𝑁 ) , the absolute values of the stability abscissas of 𝐴 𝑖 𝑗 ( f o r a l l 𝑖 , 𝑗 ∈ N ) and norms of ‖ 𝑥 ( 𝑡 ) ‖ 2 ≤ 𝑡 𝑖 ∈ S T 𝜎 ( 𝑡 ) 𝐾 𝑡 0 𝜎 𝑖 𝑒 1 + 𝜌 𝑡 𝑖 0 𝜎 ℎ − 1 𝜌 𝑡 0 𝜎 𝑖 𝑞 𝑖 = 1 ‖ ‖ 𝐴 𝑡 𝑖 𝜎 𝑖 ‖ ‖ 2 0 𝑥 0 2 0 0 𝑑 × 𝑒 − ( 𝜌 𝑖 ) 0 𝜎 ( 𝑡 − 𝐾 𝑖 ) 0 𝜎 ( 𝑡 ∑ 𝑞 𝑖 = 1 | | 𝐴 𝑖 ) 𝑖 𝜎 ( 𝑡 | | 2 ) 𝑡 𝑖 + 1 − 𝑡 𝑖 s u p − ℎ ≤ 𝜏 ≤ 0 ‖ 𝜑 ( 𝜏 ) ‖ 2 , ( 2 . 1 8 ) .

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 0 𝑗 ) < 0 is of exponential order with decay rate 𝜌 0 𝑗 > 𝐾 0 𝜎 ( 𝑡 𝑖 ) ∑ 𝑞 𝑖 = 1 ‖ 𝐴 𝑖 𝑗 ‖ 2 , f o r a l l 𝑗 ∈ 𝑁 , provided that m a x 𝑡 𝑖 ∈ S T 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ) ≥ 𝑇 , provided that the minimum residence time 𝐀 0 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 𝐀 0 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 0 𝑗 , f o r a l l 𝑗 ∈ 𝑁 are unstable with no stable or critically stable eigenvalues (i.e., all the matrices − ( 𝐴 0 𝑗 ) , f o r a l l 𝑗 ∈ 𝑁 , are antistable) and the absolute convergence abscissas of 𝜎 ∶ 𝐑 0 + → 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 ∃ l i m 𝑡 → ∞ 𝑡 𝑖 ∈ 𝑆 𝑇 𝜎 ( 𝑡 ) | | | | 𝐾 𝑡 0 𝜎 𝑖 − K 𝑡 0 𝜎 𝑖 𝑒 𝜌 𝑖 ) 0 𝜎 ( 𝑡 ℎ − 1 𝜌 𝑡 0 𝜎 𝑖 | | | | 𝑞 𝑖 = 1 ‖ ‖ A 𝑖 𝜎 ( 𝑡 𝑖 ) ‖ ‖ 2 × 𝑒 ( | ̃ 𝜌 𝑖 ) 0 𝜎 ( 𝑡 | − 𝐾 𝑖 ) 0 𝜎 ( 𝑡 ∑ 𝑞 𝑖 = 1 | | 𝐴 𝑖 ) 𝑖 𝜎 ( 𝑡 | | 2 ) 𝑡 𝑖 + 1 − 𝑡 𝑖 = ∞ , ( 2 . 1 9 ) is such that where 𝐑 − ϶ ̃ 𝜌 0 𝜎 ( 𝑡 𝑖 ) ∈ { ̃ 𝜌 0 1 , ̃ 𝜌 0 2 , … , ̃ 𝜌 0 𝑁 } and | ̃ 𝜌 0 𝑗 | ≤ | 𝜌 0 𝑗 | , with 𝐾 0 𝑗 ≤ 𝐾 0 𝑗 (with ̃ 𝜌 0 𝑗 and 𝐴 0 𝑗 being located or close to the minimum real part of the eigenvalues of 𝜌 0 𝑗 and 𝑗 ∈ 𝑁 , for all 𝜎 ( 𝑡 𝑖 ) = 𝑗 ∈ 𝑁 , defined in Theorem 2.1) if ‖ 𝑒 𝐴 0 𝑖 𝑡 ‖ 2 ≥ 𝐾 0 𝑖 𝑒 | ̃ 𝜌 0 i | 𝑡 , f o r a l l 𝑖 ∈ N are real constants such that 𝐀 0 (i.e., all the matrices in the set 𝑆 𝑇 𝜎 ( 𝑡 ) ∶ = { 𝑡 𝑖 ∈ 𝑆 𝑇 𝜎 ∶ 𝑡 𝑖 ≤ 𝑡 } are antistable and then unstable) with 𝑡 𝑠 ( 𝑡 ) + 1 ∶ = 𝑡 and 𝑡 ∉ 𝑆 𝑇 𝜎 if 𝑠 ( 𝑡 ) ∶ = c a r d 𝑆 𝑇 𝜎 ( 𝑡 ) , where 𝜎 ∶ 𝐑 0 + → 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 0 𝑗 is such that | ̃ 𝜌 0 𝑗 | > 𝐾 0 𝜎 ( 𝑡 𝑖 ) ∑ 𝑞 𝑖 = 1 ‖ 𝐴 𝑖 𝑗 ‖ 2 f o r a l l 𝑗 ∈ 𝑁 are all unstable matrices satisfying m a x 𝑡 𝑖 ∈ 𝑆 𝑇 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ) ≥ 𝑇 , and the residence time at each switching instant satisfies 𝑇 with its lower-bound | 𝜌 0 𝑗 | being sufficiently large according to the respective absolute values ( − 𝐴 0 𝑗 ) of the stability abscissas of the stable matrices . (i.e − | 𝜌 0 𝑗 | < 0 , 𝐴 0 𝑗 if all the eigenvalues of − | ̃ 𝜌 0 𝑗 | + 𝜀 ≤ − | 𝜌 0 𝑗 | + 𝜀 , 𝜀 → 0 + are distinct of positive real parts and 𝑗 ∈ 𝑁 , otherwise), for all 𝐴 𝑖 𝑗 ( f o r a l l 𝑖 , 𝑗 ∈ 𝑁 ) and norms of 𝜎 ∶ 𝐑 0 + → 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 0 𝑗 is such that at least one | 𝜌 0 𝑗 | > 𝐾 0 𝜎 ( 𝑡 𝑖 ) ∑ 𝑞 𝑖 = 1 ‖ 𝐴 𝑖 𝑗 ‖ 2 is a stable matrix satisfying m a x 𝑡 𝑖 , 𝑡 𝑖 + 1 ∈ 𝑆 𝑇 𝜎 ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ∶ 𝜎 ( 𝑡 𝑖 ) = 𝑗 ) , and furthermore, ∑ 𝑡 𝑖 , 𝑡 𝑖 + 1 ∈ 𝑆 𝑇 𝜎 m a x ( 𝑡 𝑖 + 1 − 𝑡 𝑖 ∶ 𝜎 ( 𝑡 𝑖 ) ≠ 𝑗 , 𝜎 ( 𝑡 𝑖 + 1 ) ≠ 𝑗 ) is sufficiently large compared to 𝐾 0 𝑗 ( f o r a l l 𝑗 ∈ 𝑁 ) , according to the constants 𝐴 0 𝑘 ( f o r a l l 𝑘 ∈ 𝑁 ) , the absolute values of the stability abscissas of 𝐴 𝑖 𝑗 ( f o r a l l 𝑖 , 𝑗 ∈ 𝑁 ) and norms of 𝐴 0 𝑗 . If there is only a stable matrix 𝐀 0 in the set 𝐴 0 𝑗 . If there is a unique stable matrix 𝑗 ∈ 𝑁 , for some ∑ t k , 𝑡 𝑘 + 1 ∈ 𝑆 𝑇 𝜎 ( 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 0