Explicit delay-dependent stability criteria for nonlinear distributed parameter systems

Abstract

This chapter is devoted to the stability of nonlinear autonomous systems with distributed parameters and delay, governed by functional-differential equations in a Banach space with nonlinear causal mappings and bounded operators acting on the delayed state. These equations include partial differential, integro-differential and other traditional equations. Estimates for the norms of solutions are established. They give us explicit conditions for the delay-dependent Lyapunov and exponential stabilities of the considered systems. These conditions are formulated in terms of the spectra of the operator coefficients of the equations. In addition, the obtained solution estimates provide us bounds for the regions of attraction of steady states. The global exponential stability conditions are also derived. As particular cases we consider systems with discrete and distributed delays. The illustrative examples with the Dirichlet and Neumann boundary conditions are also presented. These examples show that the obtained stability conditions allow us to avoid the construction of the Lyapunov type functionals in appropriate situations. Our approach is based on a combined usage of properties of operator semigroups with estimates for fundamental solutions of the considered equations