Conditions for stability and stabilization of systems of neutral type with nonmonotone Lyapunov functionals

Abstract

We suggest new tests for the stability and uniform asymptotic stability of an equilibrium in systems of neutral type. By using these tests, we prove conditions for optimal stabilization and derive new estimates for perturbations that can be countered by a system closed by an optimal control. We show that, by using nonmonotone sign-indefinite functionals as Lyapunov functionals, one can obtain conditions for uniform asymptotic stability that do not contain the a priori requirement of stability of the difference operator and do not imply the boundedness of the right-hand side of the system. When studying the action of perturbations on the stabilized systems, these conditions permit one to obtain new estimates of perturbations preserving the stabilizing properties of optimal controls. The obtained estimates do not imply any constraint on the value of perturbations in some domains of the phase space that are defined when constructing an optimal stabilizing control. Some examples are considered.