论文标题
抛物线PDE的最终状态，开环控制，具有Dirichlet边界条件
Final-state, Open-loop Control of Parabolic PDEs with Dirichlet Boundary Conditions
论文作者
论文摘要
在本文中，考虑了具有均匀Dirichlet边界条件的二阶抛物线PDE的二次最佳控制问题，其中“点”控制函数（仅取决于时间）构成源项。这些问题涉及选择控制函数（具有或没有“峰值”约束）以将所讨论的PDE解决方案近似转移到所需的（有限）时间间隙的结束时。为了计算所需的最佳控制函数的近似值，引入了半混凝土的半混凝土，盖尔金近似值，并解决了相应的（近似）控制问题。结果表明，解决方案的序列分别对涉及原始初始/边界价值问题的控制问题的最佳解决方案分别收敛到受约束和不受约束的近似（有限维）控制问题的序列。可以非常明确地表征对无约束近似问题的解决方案，其计算的主要数值步骤仅需要lyapunov方程的解决方案。尽管可以根据拉格朗日二元性和分段线性乘数来获得约束控制问题的近似解决方案。这些点详细研究，并通过涉及热方程（HEQ）的数值示例进行了说明。
In this paper, a quadratic optimal control problem is considered for second-order parabolic PDEs with homogeneous Dirichlet boundary conditions, in which the "point" control function (depending only on time) constitutes a source term. These problems involve choosing a control function (with or without "peak-value" constraints) to approximately steer the solution of the PDE in question to a desired function at the end of a prescribed (finite) time-interval. To compute approximations to the desired optimal control functions, semi-discrete, Galerkin approximations to the equation involved are introduced and the corresponding (approximating) control problems are tackled. It is shown that the sequences of solutions to both the constrained and unconstrained approximating (finite-dimensional) control problems converge, respectively, to the optimal solutions of the control problems involving the original initial/boundary value problem. The solution to the unconstrained approximating problem can be quite explicitly characterized, with the main numerical step for its computation requiring only the solution of a Lyapunov equation. Whereas approximate solutions to the constrained control problems can be obtained on the basis of Lagrangian duality and piecewise linear multipliers. These points are worked out in detail and illustrated by numerical examples involving the heat equation (HEq).
