非线性椭圆和抛物型变分不等式的最优控制

变分不等式是偏微分方程的重要研究分支，它在应用数学、力学、运筹学、经济学、工程科学中具有极其广泛的应用。本课题研究椭圆和抛物型变分不等式的若干最优控制问题，其主要特征如下：(1)受控状态系统是拟线性椭圆型和抛物型变分、H-半变分不等式；(2)目标函数为非光滑、非凸泛函；(3)控制变量为源函数、障碍函数或者边界函数。由于状态系统由非线性不等式所刻画并且罚函数非凸、非光滑，它们是典型的非凸、非光滑最优控制问题。本课题拟结合非光滑分析、变分—H-半变分不等式、非线性分析、偏微分方程的最优控制等理论，发展非凸、非光滑最优控制问题的研究方法和技术，研究上述最优控制问题最优解的存在性，推导最优解的一阶必要最优条件，建立最优系统。

As a significant research branch of partial differential equations, variational inequality has wide applications in applied mathematics, mechanics, operational research and engineering sciences. In this project, we aim to study some optimal control problems governed by elliptic and parabolic variational inequalities. Their key features are as follows. First of all, the state systems are described by nonlinear elliptic and parabolic variational-hemivariational inequalities. Secondly, the cost functionals of the control problems are nonconvex and nonsmooth. Finally, the control variables are chosen as the source, obstacle or boundary functions. Since the state systems are governed by nonlinear inequalities and the cost functionals are nonsmooth and nonconvex, they are typical nonconvex and nonsmooth optimal control problems. By using the theories of nonsmooth analysis, variational-hemivariational inequalities, nonlinear analysis and optimal control of partial differential equations, and developing methods and techniques for noncovex and nonsmooth optimal control problems, we aim to study the existence of optimal solutions to the aforementioned optimal control problems, derive the corresponding necessary optimality conditions of first order and establish the optimality systems.