带无界算子的半线性演化方程的最优控制的存在性

Pub Date : 2024-06-13 DOI:10.1134/s0965542524700362

A. V. Chernov

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引用次数: 0

摘要

摘要研究了希尔伯特空间中一个抽象的一阶半线性微分方程的最优控制问题，该方程具有一个无界算子，控制与右侧线性相关。假定成本函数在状态和控制方面是加法分离的，与状态有相当普遍的依赖关系。对于这个问题，证明了最优控制的存在，并建立了最优控制集的属性。作者之前关于唯一全局可解性（全局可解性）和解估计的成果在所研究方程的非线性背景下得到了发展。所指出的估计值对本研究非常重要。本研究以非线性热方程和非线性波方程为例。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Existence of an Optimal Control for a Semilinear Evolution Equation with Unbounded Operator

Abstract

An optimal control problem is investigated for an abstract semilinear differential equation of the first order in time in a Hilbert space with an unbounded operator and control involved linearly in the right-hand side. The cost functional is assumed to be additively separated with respect to state and control, with a rather general dependence on the state. For this problem, the existence of an optimal control is proved and the properties of the set of optimal controls are established. The author’s previous results on the total preservation of unique global solvability (totally global solvability) and on solution estimation for such equations are developed in the context of the nonlinearity of the equation under study. The indicated estimate is found important for the present study. A nonlinear heat equation and a nonlinear wave equation are considered as examples.

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