2022/09/26 by Landry Djomegne, Djomegne, Landry, Cyrille Kenne +5 · 2 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2209.12450
openalex publication_date 2022/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.