Hierarchical control for the semilinear parabolic equations with interior degeneracy

This paper concerns with the hierarchical control of the semilinear parabolic equations with interior degeneracy. By a Stackelberg-Nash strategy, we consider the linear and semilinear system with one leader and two followers. First, for any given leader, we analyze a Nash equilibrium corresponding to a bi-objective optimal control problem. The existence and uniqueness of the Nash equilibrium is proved, and its characterization is given. Then, we find a leader satisfying the null controllability problem. The key is to establish a new Carleman estimate for a coupled degenerate parabolic system with interior degeneracy

Hierarchic control for the coupled fourth order parabolic equations

In this paper, we obtain a null controllability result for a coupled fourth order parabolic system based on the Stackelberg-Nash strategies. For this purpose, we first prove the existence and uniqueness of Nash equilibrium pair of the original system and its explicit expression is provided. Next, we investigate the null controllability of Nash equilibrium to the corresponding optimal system. By duality theory, we establish an observability estimate for the coupled fourth order parabolic system. Such an estimate is obtained by a new global Carleman estimate we derived

Estimativas de Carleman para uma classe de problemas parabólicos degenerados e aplicações à controlabilidade multi-objetivo.

Neste trabalho apresentamos estimativas de Carleman para uma classe de problemas parabólicos degenerados sobre um quadrado (no caso bidimensional) ou sobre um intervalo limitado (no caso unidimensional). Consideramos um operador diferencial que degenera apenas em uma parte da fronteira. Provamos resultados de existência, unicidade e estimativas de energia via teoria do semigrupo. Em seguida usamos funções peso adequadas para obter estimativas de Carleman e, como aplicações, resultados de controlabilidade multi-objetivo.This work presents Carleman estimates to a class of degenerate parabolic problems over a square (in the two dimensional case) or a bounded interval (in the one dimensional case). We consider a di erential operator that degenerate only in a part of the boundary. Using semigroup theory, we prove well posedness results. Then, using suitables weight functions, we prove Carleman estimates and, as application, results on multi-objective controllability.Cape

Carleman inequality for a class of super strong degenerate parabolic operators and applications

In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in general. Additionally, we also apply the aforementioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem

Carleman inequality for a class of super strong degenerate parabolic operators and applications

In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in the general situations. Additionally, we also apply the aformentioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem

On Pareto equilibria for bi-objective diffusive optimal control problems

We investigate Pareto equilibria for bi-objective optimal control problems. Our framework comprises the situation in which an agent acts with a distributed control in a portion of a given domain, and aims to achieve two distinct (possibly conflicting) targets. We analyze systems governed by linear and semilinear heat equations and also systems with multiplicative controls. We develop numerical methods relying on a combination of finite elements and finite differences. We illustrate the computational methods we develop via numerous experiments