Hierarchical Control for the Wave Equation with a Moving Boundary
Isaías Pereira de Jesus
TL;DR
The paper studies hierarchical (Stackelberg-Nash) control of the one-dimensional wave equation on a non-cylindrical domain with a moving boundary, driven by a boundary leader on $\\widehat{\\Sigma}_1$ and a boundary follower on $\\widehat{\\Sigma}_2$, under the moving-front speed constraint $0<k<1$ with boundary position $\\alpha_k(t)=1+kt$. By transforming to a fixed cylinder and formulating two costs, it proves the existence and uniqueness of a Nash equilibrium for the follower given the leader, characterizes the equilibrium via an adjoint-based optimality system, and demonstrates approximate controllability with respect to the leader under a time-condition $T>T_k^*$. It further derives a leader-optimality system using Fenchel–Rockafellar duality, yielding an explicit leader control in terms of the adjoint variable, $w_1=-\frac{1}{\\alpha_k^2(t)}\\varphi_y$ on the leader boundary. The results advance hierarchical control for PDEs with time-dependent domains and provide a framework for leader-driven approximate steering of wave dynamics with moving boundaries.
Abstract
This paper addresses the study of the hierarchical control for the one-dimensional wave equation in intervals with a moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We assume that we can act on the dynamic of the system by a hierarchy of controls. According to the formulation given by H. von Stackelberg (Marktform und Gleichgewicht, Springer, Berlin, 1934), there are local controls called followers and global controls called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as ``the leader" and the other one as ``the follower". This situation is studied in the paper, with one of the cost functions being of the controllability type. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.