arXiv subjects

Pascal Zongo

Publications and source records attributed to Pascal Zongo.

3 recordsLinked to original sources

arXiv2022

Second-order optimality conditions for the bilinear optimal control of a degenerate equation

The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system.

math.OC↗

arXiv2022

Stackelberg-Nash null controllability for a non linear coupled degenerate parabolic equations

The main purpose of this paper is to apply the notion of hierarchical control to a coupled degenerate non linear parabolic equations. We use the Stackelberg-Nash strategy with one leader and two followers. The followers solve a Nash equilibrium corresponding to a bi-objective optimal control problem and the leader a null controllability problem. Since the considered problem is non linear, the associated cost is non-convex. We first prove the existence, uniqueness and the characterization of the Nash quasiequilibrium, which is a weak formulation of the Nash equilibrium because the cost associated to the non linear problem is non-convex. Next, we show that under suitable conditions, the Nash quasi-equilibrium is equivalent to the Nash equilibrium. Finally using some Carleman inequalities that we established, and the Kakutani's fixed point Theorem, we brough the states of our system to the rest at final time T .

math.CO↗

arXiv2022

Hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term

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.

math.OC↗