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

and Applied Analysis 3 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 timeinvariant 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 and R denote the positive subsets of Z, respectively, and C denotes the subset of C of complex numbers with positive real part. Z− and R− denote the negative subsets of Z, respectively, and C− denotes the subset of C of complex numbers with negative real part. Z0 : Z ∪ {0}, R0 : R ∪ {0}, C0 : C ∪ {0}, Z0− : Z− ∪ {0}, R0− : R− ∪ {0}, C0− : C− ∪ {0}. 1.1 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 4 Abstract and Applied Analysis 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 L2 ≡ L2 R and endowed with the inner product L2-norm ‖f‖Ln2 : ∫∞ −∞‖f τ ‖2dτ , for all f ∈ Ln2 , where ‖ · ‖2 is the 2-vector or Euclidean norm and its corresponding induced matrix norm. Ln2 α,∞ the Hilbert space of n norm-square Lebesgue integrable real functions on α,∞ ⊂ R for a given α ∈ R which is endowed with the norm ‖f‖Ln2 α,∞ : ∫∞ α ‖f τ ‖2dτ , for all f ∈ L2 α,∞ . L2 : {f ∈ L2 : f t 0, for all t ∈ R−} and Ln2− : {f ∈ Ln2 : f t 0, for all t ∈ R } are closed subspaces of L2 : {f ∈ Ln2 : f t 0, for all t ∈ R−} → L2 of respective supports R0 and R0−. Then, Ln2 Ln2 ⊕ Ln2−. In denotes the nth identity matrix. λmax M and λmin M stand for the maximum and minimum eigenvalues of a definite square real matrix M mij . 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. στ,t : σ | 0, t : 0, t ⊂ R0 → Nτ,t ⊂ N is the partial switching function with its domain restricted to τ, t . σt is a notational abbreviation of σ0,t. The point constant delays are denoted by hi ∈ 0, h , for all i ∈ 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 < · · · < hq h: