Componentwise Asymptotic Stability Induced by Symmetrical Polyhedral Time-Dependent Constraints

Abstract

In this paper the concepts of componentwise asymptotic stability with respect to a differentiable vector function h( t) (approaching 0 as t -+ 00) (CWASh) and componentwise exponentially asymptotic stability (CWEAS), previously introduced, have been extended to Q-CWASh and Q-CWEAS (Q being a q x n real matrix), respectively, in order to cover the more general situation of polyhedral time-dependent fiowinvariant sets, defined by JQxJ :::; h(t), x E IR n , t E 1R+, symmetrical with respect to the equilibrium point of a given continuous-time linear systemwith the existence of a q x q matrix E such that EQ = QA, Eh(t) :::; h(t), where the bar operator C) transforms only the extra diagonal elements of E into their corresponding absolute values and does not change its diagonal elements. By specializing vector function h(t) in an exponentially decaying form, the concept of Q-CWEAS is characterized by the above mentioned matrix equation and an algebraic inequality. For Q = In these results consistently yield the earlier ones. As in this case, there exists a strong connection between Q-CWASh (Q-CWEAS) and the asymptotic stability, but now this connection is amended by the observability of the pair (Q, A).