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.

在这项工作中，我们建立了有限时滞中性线性积分微分方程的基本解理论，包括存在性和一些基本性质。这样，我们就可以通过拉普拉斯变换方法用基本解来表示半线性中性延迟积分-微分方程的温和解。然后，作为应用，我们基于基本解、解算子和分数幂理论研究了这类积分-微分进化系统的近似可控性、最优控制和时间最优控制问题，这些控制问题的充分条件是分别获得。最后，给出一个例子来说明所取得的结果。