Stabilization of linear continuous time-varying systems with state delays in Hilbert spaces

This paper studies the stabilization of the infinite-dimensional linear time-varying system with state delays ẋ = A(t)x+A1(t)x(t− h) +B(t)u . The operator A(t) is assumed to be the generator of a strong evolution operator. In contrast to the previous results, the stabilizability conditions are obtained via solving a Riccati differential equation and do not involve any stability property of the evolution operator. Our conditions are easy to construct and to verify. We provide a step-by-step procedure for finding feedback controllers and state stability conditions for some linear delay control systems with nonlinear perturbations.