Our main aims in this paper are to investigate the regularity of inertial manifolds for non-autonomous semi-linear evolution equations and to give an application of inertial manifolds to a feedback control problem. We first prove that the inertial manifolds are smooth if the nonlinear term is smooth. Then, using the theory of inertial manifolds for non-autonomous semi-linear evolution equations, we construct a feedback controller for a class of control problems for the one-dimensional reaction-diffusion equations with the Lipschitz coefficient of the nonlinear term which may depend on time and belong to an admissible space.

中文翻译：

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容许空间中的演化方程和有限维反馈控制器的惯性流形的正则性

本文的主要目的是研究非自治半线性发展方程的惯性流形的规律性，并将惯性流形应用于反馈控制问题。我们首先证明，如果非线性项是光滑的，则惯性流形是光滑的。然后，使用惯性流形理论求解非自治半线性发展方程，我们针对一维反应扩散方程的非线性项的Lipschitz系数构造了一类控制问题的反馈控制器，该非线性项的Lipschitz系数可能取决于时间并属于一个允许的空间。