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双线性系统作为一类特殊非线性系统，广泛应用于工程，生物，经济等诸多领域，在实际应用和理论研究中占据重要地位，其理论发展历经多年，从早期理论框架的构建到各类原理、方法的提出，不断丰富完善。本文重点聚焦偏微分方程约束下的双线性最优控制问题，深入探讨椭圆方程、抛物方程、对流-扩散方程等不同类型偏微分方程约束双线性最优控制问题的求解策略，系统综述有限元方法、多重网格算法与自适应算法等多种数值求解手段。同时，本文阐述近年来偏微分方程约束双线性最优控制问题在多学科交叉应用中的新成果。最后，本文对未来研究方向予以展望，以期推动双线性最优控制领域的进一步发展。
Abstract：Bilinear systems, as a special class of nonlinear systems, find widespread applications in diverse fields such as engineering, biology, and economics, where they play a significant role in both practical applications and theoretical research. Over the years, the theory of bilinear systems has continuously evolved, progressing from the establishment of an initial theoretical framework to the development of various principles and methodologies, thereby becoming increasingly comprehensive and sophisticated. This paper focuses on bilinear optimal control problems governed by partial differential equations(PDEs). It thoroughly investigates solution strategies for such problems involving different types of PDEs, including elliptic, parabolic, and advection-diffusion equations. Furthermore, it provides a systematic review of various numerical solution methods commonly employed in this field, such as the finite element method, the multigrid algorithm, and adaptive algorithms. At the same time, it expounds on the new achievements in the interdisciplinary applications of bilinear optimal control problems governed by PDEs in recent years. Finally, it outlines potential future research directions aimed at promoting further advancements in the field of bilinear optimal control.
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基本信息:
DOI：
中图分类号:O232;O175.2
引用信息:
[1]王同昕,周兆杰.偏微分方程约束双线性最优控制问题的研究进展[J].山东师范大学学报(自然科学版),2025,40(03):219-231.
基金信息:
国家自然科学基金资助项目(11971276)