This paper is devoted to a studying of the controllability properties for the Coleman–Gurtin-type equation, which is a class of multidimensional integral–differential equations. The goal is to prove the existence of a control function which steers the state variable and the integral term to the neighborhood of two given final configurations at the same time, respectively. This new approximate controllability is defined by imposing some additional integral-type constraints on the usual approximate controllability, ensuring that the whole process reaches the neighborhood of the equilibrium. We also provide a characterization of the initial values, which can be driven to zero by a distributed control. The later is a supplement of non-null controllability for the Coleman–Gurtin model in the square integrable space.

本文致力于研究Coleman–Gurtin型方程的可控性，该方程是一类多维积分－微分方程。目的是证明控制功能的存在，该控制功能可同时将状态变量和积分项导向两个给定的最终配置的邻域。这种新的近似可控性是通过在通常的近似可控性上施加一些附加的整数约束来定义的，从而确保整个过程都达到平衡点附近。我们还提供了初始值的特征，可以通过分布式控制将其驱动为零。后者是正方形可积空间中Coleman-Gurtin模型非零可控性的补充。