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In this work, we establish the theory of fundamental solution, including the existence and some essential properties, for neutral linear integro-differential equations with finite delay. In this way, we are able to represent the mild solutions of semi-linear neutral retarded integro-differential equations by fundamental solutions through the Laplace transformation method. Then, as applications, we investigate the problems of approximate controllability, optimal control and time optimal control for this kind of integro-differential evolution systems based on the theory of fundamental solution, resolvent operators and fractional powers, and sufficient conditions for these control problems are obtained, respectively. Finally, an example is presented to illustrate the achieved results.

Fundamental solutions for semi-linear neutral retarded integro-differential systems and applications to control problems