论文标题
分数扩散方程的双线性最佳控制
Bilinear optimal control for a fractional diffusive equation
论文作者
论文摘要
我们考虑涉及订单$ 0 <s <1 $的分数拉普拉斯操作员的演变方程的双线性最佳控制。我们首先为所考虑的演化方程提供了一些存在和独特性结果。接下来，我们建立了一些较弱的最大原则结果，使我们能够获得更多的状态方程式。然后，我们考虑一个最佳控制问题，该问题将最终系统状态带到所需状态。我们表明，这个最佳控制问题具有解决方案，我们得出了第一阶和二阶最佳条件。最后，在初始基准和给定目标的其他假设下，我们证明可以实现最佳解决方案的局部唯一性。
We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order $0<s<1$. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first and second order optimality conditions. Finally, under additional assumptions on the initial datum and the given target, we prove that local uniqueness of optimal solutions can be achieved.
