论文标题
第四阶抛物线方程的分层精确可控性
Hierarchical exact controllability of the fourth order parabolic equations
论文作者
论文摘要
本文涉及使用Stackelberg-Nash策略来控制第四阶线性和半线性抛物线方程。我们假设该系统是通过分布式控件的层次结构来起作用的：负责确切可控性属性的主要控制（领导者）；还有几个辅助控件（追随者），可最大程度地减少两个规定的成本功能，并为两个规定的成本功能提供一对NASH均衡。在本文中，我们首先证明了相关的NASH平衡对的存在，该对应于Banach固定点定理对每个领导者的分层双目标最佳控制问题。然后，我们通过全球卡尔曼不平等和能量方法建立了第四阶耦合抛物线方程的可观察性不平等。基于这种结果，我们获得了将受控系统精确地驱动到规定（但任意）轨迹的领导者的存在。此外，我们还为二级控件提供了二阶的最佳条件。
This paper is concerned with the application of Stackelberg-Nash strategies to control fourth order linear and semi-linear parabolic equations. We assume that the system is acted through a hierarchy of distributed controls: one main control (the leader) that is responsible for an exact controllability property; and a couple of secondary controls (the followers) that minimize two prescribed cost functionals and provides a pair of Nash equilibria for the two prescribed cost functionals. In this paper, we first prove the existence of an associated Nash equilibrium pair corresponding to a hierarchical bi-objective optimal control problem for each leader by Banach fixed points theorem. Then, we establish an observability inequalities of fourth order coupled parabolic equations by global Carleman inequalities and energy methods. Based on such results, we obtain the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory. Furthermore, we also give the second-order sufficient conditions of optimality for the secondary controls.
