Sufficient conditions for robust stability of linear nonstationary control systems with periodic interval constraints
The authors consider a linear nonstationary control system of the form \[ \dot x = A(t)x,\quad x \in R^n \] where \(A(t)\) is an \(n \times n\)-matrix function satisfying a.e. on each bounded interval of the semiaxis \([0,\infty)\) the following constraints: \(\underline A(t) \leq A(t) \leq \bar A(t)\) where the given bounded measurable matrix functions \(\underline A(t)\) and \(\bar A(t)\) are assumed to be periodic. The system is said to be robust stable with respect to given periodic constraints provided its zero solution is Lyapunov asymptotically stable for any choice of a matrix function \(A(t)\) satisfying the given constraints. Sufficient conditions for robust stability are given on the base of the method of comparison with a Lyapunov vector function of a special form. It is demonstrated that these conditions are also necessary under some additional assumptions. The results are generalized to the case of linear control systems with polyhedral periodic constraints.
scientific article; zbMATH DE number 1290216