Approximate controllability of impulsive system involving state-dependent delay and variable delay in control via fundamental solution

Abstract

This article is concerned with the approximate controllability for a new class of impulsive semilinear control systems involving state-dependent delay and variable delay in control in Hilbert spaces. We formulate new sufficient conditions which guarantee the existence of solution to the considered system. We use the theory of fundamental solution, Krasnoselskii?s and Schauder?s fixed point theorems to establish our major results. Finally, two examples are constructed which demonstrate the effectiveness of obtained results.

Necessary and sufficient conditions for the approximate controllability of fractional linear systems via C−semigroups

“…the control function u is given in L 2 (J, Y), Y is a Hilbert space; B is a bounded linear operator from Y to X. We establish four necessary and sufficient conditions of approximate controllability in the resolvent form for system (1). The proof is based on the characterization of the symmetric operator as well as the duality mapping.…”

“…In section 2, we recall some definitions of Caputo fractional derivatives and C−semigroups. We also obtain the definitions of mild solutions and approximate controllability of system (1). The corresponding nonnegative and symmetric operator and duality mapping are also introduced.…”