论文标题
半线性退化抛物线方程的层次无效可控性
Hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term
论文作者
论文摘要
在本文中，我们将分层策略应用于涉及梯度项的半线性弱退化抛物线方程。我们将stackelberg-nash策略与一个领导者一起使用，该策略试图将解决方案推向零，两个追随者旨在解决与双目标最佳控制问题相对应的NASH平衡。由于系统是半线性，因此功能通常不是凸。为了克服这一困难，我们首先证明了NASH准平衡的存在和独特性，这是NASH平衡的弱公式。接下来，在其他条件下，我们建立了NASH准平衡与NASH平衡之间的等效性。我们为伴随系统建立了合适的卡尔曼不平等，然后是可观察性不平等。基于这种可观察性不等式，我们证明了线性化系统的无效可控性。然后，由于Kakutani的固定点定理，我们获得了主系统的无效可控性。
In this paper, we apply the hierarchical strategy to a semilinear weakly degenerate parabolic equation involving a gradient term. We use the Stackelberg-Nash strategy with one leader which tries to drive the solution to zero and two followers intended to solve a Nash equilibrium corresponding to a bi-objective optimal control problem. Since the system is semilinear, the functionals are not convex in general. To overcome this difficulty, we first prove the existence and uniqueness of the Nash quasi-equilibrium, which is a weaker formulation of the Nash equilibrium. Next, with additional conditions, we establish the equivalence between the Nash quasi-equilibrium and the Nash equilibrium. We establish a suitable Carleman inequality for the adjoint system and then an observability inequality. Based on this observability inequality, we prove the null controllability of the linearized system. Then, due to the Kakutani's fixed point Theorem, we obtain the null controllability of the main system.
