This paper is concerned with the structural controllability analysis for discrete-time linear control systems with time-delay. By adding virtual delay nodes, the linear systems with time-delay are transformed into corresponding linear systems without time-delay, and the structural controllability of them is equivalent. That is to say, the time-delay does not affect or change the controllability of the systems. Several examples are also presented to illustrate the theoretical results.
In this paper, we first construct strong Lyapunov functions for random attractors. On this basis, we then establish strong Morse–Lyapunov functions for Morse decompositions of the random attractors. Finally, the stability of the Morse decompositions is studied through the Morse–Lyapunov functions.
Abstract. In this paper, we show that general nonlinear switched systems are asymptotically controllable if and only if there exist controlLyapunov functions for their relaxation systems. If the switching signal is dependent on the time, then the control-Lyapunov functions are continuous. And if the switching signal is dependent on the state, then the control-Lyapunov functions are C 1 -smooth. We obtain the results from the viewpoint of control system theory. Our approach is based on the relaxation theorems of differential inclusions and the classic Lyapunov characterization.
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