论文标题
非线性耦合退化抛物线方程的Stackelberg-Nash无效可控性
Stackelberg-Nash null controllability for a non linear coupled degenerate parabolic equations
论文作者
论文摘要
本文的主要目的是将分层控制的概念应用于耦合的退化非线性抛物线方程。我们将Stackelberg-Nash策略与一个领导者和两个追随者一起使用。追随者解决了对应于双目标最佳控制问题的NASH均衡，并解决了无效的可控性问题。由于所考虑的问题是非线性的，因此相关的成本是非凸面。我们首先证明了NASH Quasiequilibrium的存在，唯一性和表征，这是NASH平衡的弱公式，因为与非线性问题相关的成本是非convex。接下来，我们表明，在适当的条件下，NASH准平衡等效于NASH平衡。最终，我们确定了一些卡尔曼（Carleman）的不平等现象，以及卡库塔尼（Kakutani）的固定点定理，我们将系统状态施加到最后一次t。
The main purpose of this paper is to apply the notion of hierarchical control to a coupled degenerate non linear parabolic equations. We use the Stackelberg-Nash strategy with one leader and two followers. The followers solve a Nash equilibrium corresponding to a bi-objective optimal control problem and the leader a null controllability problem. Since the considered problem is non linear, the associated cost is non-convex. We first prove the existence, uniqueness and the characterization of the Nash quasiequilibrium, which is a weak formulation of the Nash equilibrium because the cost associated to the non linear problem is non-convex. Next, we show that under suitable conditions, the Nash quasi-equilibrium is equivalent to the Nash equilibrium. Finally using some Carleman inequalities that we established, and the Kakutani's fixed point Theorem, we brough the states of our system to the rest at final time T .
