#### 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 -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. *, and are the sets of integer, real, and complex numbers, respectively.

denote the positive subsets of , respectively, and denotes the subset of of complex numbers with positive real part.

denote the negative subsets of , respectively, and denotes the subset of of complex numbers with negative real part.

Given some linear space (usually or ) then denotes the set of functions of class . Also, and denote the set of functions in which, furthermore, possess bounded piecewise continuous constant or, respectively, piecewise continuous constant derivative on .

The set of linear operators from the linear space to the linear space are denoted by , and the Hilbert space of norm-square Lebesgue integrable real functions on is denoted by and endowed with the inner product -norm , for all , where is the -vector (or Euclidean) norm and its corresponding induced matrix norm. the Hilbert space of norm-square Lebesgue integrable real functions on for a given which is endowed with the norm , for all . and are closed subspaces of of respective supports and . Then, .

denotes the th identity matrix.

and stand for the maximum and minimum eigenvalues of a definite square real matrix .

is the switching function which defines the parameterization at time of a switched dynamic system among possible time invariant parameterizations. is the partial switching function with its domain restricted to . is a notational abbreviation of .

The point constant delays are denoted by and are, in general, incommensurate, and .

#### 2. The Dynamic System Subject to Time Delays

Consider the following class of switched linear time-varying differential dynamic system subject to distinct internal incommensurate point delays :

where ; for all , , , and are the state, input (or control) and output (or measurement) vectors, respectively, and

where , fulfilling that , , and are piecewise constant such that they are constant either in or in , for all and some fixed . 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):; is the particular system arising when all the delays of (2.1) are zero. (ii)

*The delay-free unforced auxiliary switched system:*; 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 , which generates a strictly ordered sequence of switching time instants , and which might be equivalently rewritten, since , for all , , , via the switching function , as

where is the state-trajectory solution, which is almost everywhere time differentiable on and satisfies (2.3), subject to bounded piecewise continuous initial conditions on , that is, . It is assumed that , , being the first switching instant generated by the switching function ; that is, there is a time invariant parameterization belonging to the given set on . The above assumption has an obvious real meaning for the general cases where the control is nonzero on. The unique mild solution of the state-trajectory solution, which exists on according to Picard-Lindeloff theorem for any given and any , may be calculated on any time interval 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 ; , , for all and . Note that , for all . Then, the state trajectory solution is

where, although the evolution operators between any two time instants depends on the corresponding partial switching function , the simpler notation is preferred instead for for the sake of simplicity. This simplified notation criterion will be used when no confusion is expected together with the former one for all the matrices of the individual parameterizations. The output trajectory solution is

for all , subject to initial conditions , where

(1) is the strip of state-trajectory solution on which takes values if

(2) the evolution operator in is defined pointwisely by

so that is the unforced response in , where the matrix function is a fundamental matrix of the dynamic differential system which is everywhere differentiable and has almost everywhere continuous time-derivative on with bounded discontinuities on the set and is defined on the interval as

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

(3) the input-state and input-output operators in and , respectively, , are defined pointwisely by

where

so that

are, respectively, the unforced state and output responses in . 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 so that , is an everywhere differentiable matrix function on , with almost everywhere continuous time-derivative except at time instants in , which satisfies

on whose unique solution satisfies , for all , , and is defined by

on any time interval . Now, take , and consider that the input is defined on . Then, the combination of (2.7) with the substitution of (2.13) in the delayed state and output-trajectory solutions yields

where 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 is such that
**
where and if are real constants such that (i.e., all the matrices in the set are stable) with and if , where . **
(ii) The unforced dynamic system (2.1), (2.3) is globally exponentially stable independent of the sizes of the delays if the switching function is such that are all stable matrices satisfying , for all , and the residence time at each switching instant satisfies with its lower-bound T being sufficiently large according to the respective absolute values of the stability (or convergence) abscissas of (i.e., if all the eigenvalues of are distinct and , otherwise), for all and the norms of the matrices . **
(iii) The unforced dynamic system (2.1), (2.3) is globally exponentially stable independent of the sizes of the delays if the switching function is such that at least one is a stable matrix satisfying , and furthermore, is sufficiently large compared to , according to the constants , the absolute values of the stability abscissas of , and norms of . If there is only a stable matrix in the set . If there is a unique stable matrix , for some , then the switched system is globally exponentially stable only if the switching function is such that has infinite measure. If there is a unique stable matrix for some and if the sequence of switching instants is finite, then the switching function is such that for the last switching instant . **
(iv) If where , and are the sets of stable, unstable, and critically stable matrices in the set then the switched system is globally exponentially stable independent of the sizes of the delays if the switching function is such that is sufficiently large compared to according to the constants , the absolute values of the stability abscissas of and norms of .*

*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 as 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 is of exponential order with decay rate , provided that, provided that the minimum residence time 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 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 are unstable with no stable or critically stable eigenvalues (i.e., all the matrices , are antistable) and the absolute convergence abscissas of , 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 is such that
**
where and , with (with and being located or close to the minimum real part of the eigenvalues of and , for all , defined in Theorem 2.1) if are real constants such that (i.e., all the matrices in the set are antistable and then unstable) with and if , where . **
(ii) The unforced dynamic system (2.1), (2.3) is globally exponentially unstable independent of the sizes of the delays if the switching function is such that are all unstable matrices satisfying , and the residence time at each switching instant satisfies with its lower-bound being sufficiently large according to the respective absolute values of the stability abscissas of the stable matrices (i.e, if all the eigenvalues of are distinct of positive real parts and , otherwise), for all and norms of . **
(iii) The unforced dynamic system (2.1), (2.3) is globally exponentially unstable independent of the sizes of the delays if the switching function is such that at least one is a stable matrix satisfying, and furthermore, is sufficiently large compared to , according to the constants , the absolute values of the stability abscissas of and norms of . If there is only a stable matrix in the set . If there is a unique stable matrix , for some , then the switched system is globally exponentially stable only if the switching function is such that has infinite measure. If there is a unique stable matrix for some and if the sequence of switching instants is finite, then the switched system is globally exponentially stable if the switching function is such that for the last switching instant .*

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 and , respectively. As a result, the input-state and input-output operators are members of and , 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 then the unforced state and output trajectory solutions and , for all , respectively. Thus, and which are then linear bounded operators since the switched system is either globally asymptotically stable or globally exponentially stable. In particular, and . **If, in addition, for some then the respective forced solutions fulfil and , for all , which are then bounded operators. Thus, and . **If then the respective forced solutions fulfil and , respectively, so that and . Equivalently, if , that is, , for all , then and . Equivalently, if , then and . **If Theorem 2.2 holds and , then the respective forced solutions fulfil and so that and . Equivalently, if , then and . **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 on its appropriate definition domains.*If the system is globally asymptotically stable, then it is possible to restrict the domain and image and of to and , respectively, for vector functions such that , since their support is and , and then to define a restricted operator . In the same way, it is possible to define a restricted operator . Similarly, it is possible to define and of usefulness for vector functions if the system is unstable.*

#### 3. Input-State and Input-to-Output Operators of the Switched System and Hankel and Toeplitz Operators

This section investigates the input-state and input-output operators and of the switched system (2.1), and explicit expressions defining them are given. Then, if the input is a square-integrable real -vector on , further conditions for and are investigated and weaker ones are also given for or with being a bounded real interval, in particular for . Finally, The Hankel and Toeplitz causal and anticausal operators are investigated concerning the cases . 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. *, and the matrices are dichotomic (i.e., they have no eigenvalues on the imaginary axis) while they have stable and antistable diagonal blocks and of the same respective orders and , for all , which satisfy . Furthermore, the norms of all the matrices of delayed dynamics are less than so that Theorem 2.1 (resp., Theorem 2.2) holds if all the matrices in the set are stable (resp., antistable).

*Assumption 3.2. *Assumption 3.1 holds and, furthermore, the matrices are simultaneously block diagonalizable through the same transformation matrix; for all .

Note that if Assumption 3.1 hold then no configuration of the switched system has eigenvalues within the open vertical strip of the complex plane from Theorems 2.1 and 2.2. Furthermore, there exist nonunique coordinate transformations , for all , such that

where is stable (i.e., all its eigenvalues are in ) and of order , and is antistable (i.e., all its eigenvalues are in ) and of order , for all . Note also that if Assumption 3.2 holds, then , for all . After a linear change of variables , for all with , such that and , for some , the system (2.1) may be described as follows: for all , where

for some such that subject to (3.1) and (3.4) below: for all and some for all where

The subspaces and are independent of and are called, respectively, the stable and antistable subspaces of , for all , which are complementary, that is, , for all , so that , for all . The projections on those subspaces are given by the respective formulas:

and for some such that ; for all for each . Thus, from (2.13)–(2.15), and , one gets directly

with initial conditions , so that , is an everywhere differentiable matrix function on , with almost everywhere continuous time-derivative except at time instants in , which satisfies:

on , since , for all , whose unique solution satisfies , for all , and is defined by

Then, so that

for all , with , for all , for all since

for all , for all provided that , 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 . 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 .

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 may be or not to be a switching instant and that the transformation of variables are given by a nonsingular matrix which takes a finite number of values and which is constant within the semiopen time interval in-between any two consecutive switching instants:

where 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 , , , for .

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)
dependent on the switching instants on (i.e., depends on the partial switching function ) such that
**
which is nonsingular for any finite arguments irrespective of Assumption 3.1, where is the partial switching function with its domain restricted to . and are, in general, of time interval-dependent sizes , 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 , . **
(iii) If Assumption 3.2 holds, then (ii) holds with constant , for all . **
(iv) If Assumption 3.2 holds, and all the matrices in the set defining the switched system by the partial switching function up to time t defined as commute, so that , then
**
for all , for all . 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 defining the switched system by the partial switching function up to time have a block diagonal structure with two block matrices of common sizes and , then is block diagonalizable with two nonzero square matrix blocks of time invariant sizes and , for all . 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 is an exponential matrix function of within interswitching time intervals which is block diagonalizable under the same similarity transformation and with the same block diagonal matrices sizes as the matrix, the stable (antistable) block diagonal matrix generating a convergent (divergent) exponential matrix function .

(iv) Its first part follows from (2.9) since for any real constants , and any which commute, , for all . Its second part follows from the semigroup property of .

(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, and , under the given assumptions since the matrices , for all , are diagonalizable with identical two square matrix blocks of identical sizes.

If all the matrices in the set 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 the usual orthogonal projections of onto. 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. * and are linear bounded, equivalently continuous, operators if any of the properties Theorem 2.1(i)–(iii) holds, and and 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 ) and, furthermore, square-integrable, then the state and output trajectory solutions are identically zero for and square-integrable on . 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 and , the input-state operators and input-output (identified with the so-called causal Toeplitz operator if the input is an original vector function) have domains restricted from to and projected images from , respectively, , onto , respectively,, provided that and . In the same way, the input-state operators and input-output have domains restricted from to and projected images from , respectively, , onto , respectively, , provided that and . 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 and subject to domain restrictions and orthogonal projections of their images since they act on half axis Lebesgue spaces .

*Definition 3.7. *Let be bounded, so that is also bounded. We define the following:(1)the causal input-output Hankel operator (or, simply causal Hankel operator) with symbol ,(2)the anticausal input-output Hankel operator (or, simply anticausal Hankel operator) with symbol ,(3)the causal input-output Toeplitz operator (or, simply causal Toeplitz operator) with symbol ,(4)the anticausal input-output Toeplitz operator (or, simply anticausal Toeplitz operator) with symbol ,(5)the causal input-state Hankel operator with symbol ,(6)the anticausal input-state Hankel operator with symbol ,(7)the causal input-state Toeplitz operator with symbol ,(8)the anticausal input-state Toeplitz operator with symbol .

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)
, so that if and only if , with
**, so that if and only if , with
**
(ii)
, so that if and only if , with
**, so that if and only if , with
**
(iii)
,
**
The last expression being valid if since *