2025/09/10 by Haiyang Lin, Bo You, Lin, Haiyang +1
Computer Science · Engineering · #Adaptive Dynamic Programming Control #Analysis of PDEs (math.AP) #Control and Stability of Dynamical Systems #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2509.08471
openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main objective of this paper is to study the hierarchical exact controllability for a parabolic equation with Hardy potential by Stackelberg-Nash strategy. In linear case, we employ Lax-Milgram theorem to prove the existence of an associated Nash equilibrium pair corresponding to a bi-objective optimal control problem for each leader, which is responsible for an exact controllability property. Then the observability inequality of a coupled parabolic system is established by using global Carleman inequalities, which results in the existence of a leader that drives the controlled system exactly to any prescribed trajectory. In semilinear case, we first prove the well-posedness of the coupled parabolic system to obtain the existence of Nash quasi-equilibrium pair and show that Nash quasi-equilibrium is equivalent to Nash equilibrium. Based on these results, we establish the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory by Leray-Schauder fixed point theorem.