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On the characterization of Hankel and Toeplitz operators describing switched linear dynamic systems with point delays. (English) Zbl 1200.37019

The dynamic system under investigation is a linear switched system subjected 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 parametrization 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 the input-output operators under zero initial conditions. The causal and anticausal Toeplitz as well as 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 delay 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 [V. Ionescu, C. Dara, M. Weiss, Generalized Riccati theory and robust control. A Popov function Approach. Chichester: John Wiley & Sons. (1999; Zbl 0915.34024)]

At first the author 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 for the case when the auxiliary unforced delay-free system possessed 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. Further are discussed 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 operators 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 switching system are characterized, and some relationships between the properties of causality, stability, controllability, and observability are also proven.

MSC:
37C75Stability theory