论文标题
关于控制半连接部分偏微分方程问题的解决方案的注释
A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations
论文作者
论文摘要
在本文中，我们研究了半线性椭圆和抛物线方程的最佳控制问题。考虑了在控制和状态变量中的跟踪成本功能，二次。没有施加控制约束。我们证明，相应的状态方程在$ l^2 $中的控件方面适合。但是，众所周知，在$ l^2 $框架中，控制问题所涉及的映射通常不可差，这使得对最佳条件具有挑战性的任何分析。但是，我们证明，每个$ l^2 $最佳控制属于$ l^\ infty $，因此可以使用标准的最佳条件。
In this paper, we study optimal control problems of semilinear elliptic and parabolic equations. A tracking cost functional, quadratic in the control and state variables, is considered. No control constraints are imposed. We prove that the corresponding state equations are well-posed for controls in $L^2$. However, it is well-known that in the $L^2$ framework the mappings involved in the control problem are not Frechet differentiable in general, which makes any analysis of the optimality conditions challenging. Nevertheless, we prove that every $L^2$ optimal control belongs to $L^\infty$, and consequently standard optimality conditions are available.
