Stabilization of slowly time-varying discrete systems with state delays

This paper studies the stabilization of the infinite-dimensional linear discrete time-varying systems with state delays and slowly varying coefficients x(k + 1) = A(k)x(k) + A(1)(k)x(k - r) + B(k)u(k), k = 0,1, ... , where A(k) are bounded linear operators acting on a Banach space X, B(k) are X-valued bounded linear operators defined on a Banach space U and A(1)(k) are bounded linear operators acting on a Banach space X. Assuming appropriate conditions, we will show that the stabilization of the 'frozen' (time invariant) system x(k + 1) = A(m)x(k) + A(1)(m)x(k - r) + B(m)u(k), for each fixed integer m, implies the stabilization of the corresponding time-varying system. For such systems, explicit conditions for the feedback exponential stabilizability are established. As an application of the main results, we study the exponential stabilizability of a general nonlinear control system with several state delays.