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Nash-Stackelberg controllability for coupled systems of degenerate equations in non-cylindrical domains

Alfredo S. Gamboa, Juan Limaco, Luis P. Yapu

TL;DR

The article addresses local null controllability for a coupled, degenerate semilinear parabolic system in a time-dependent, non-cylindrical domain under a Stackelberg–Nash hierarchy with one leader and two followers. It first transforms the moving-domain problem to a cylindrical one, derives a Nash quasi-equilibrium via the linear–quadratic optimality system, and proves a Carleman inequality for the linearized adjoint system to obtain a global observability estimate. These linear results feed into Liusternik's inverse function theorem to establish local null controllability of the nonlinear system in a small-data regime, with a leader and follower controls driving both states to zero at the terminal time. The work extends hierarchical control methods to degenerate, coupled parabolic equations in moving domains, using a degenerate Carleman framework and a nonlinear inversion argument to achieve controllability.

Abstract

In this paper we investigate the Hierarchical null controllability of a coupled degenerate semilinear parabolic equation in domains which are moving in time. We show the local null controllability of the semilinear system using Liusternik's inverse function theorem. Nevertheless, the main difficulty is to adapt a Carleman estimate for the controllability of the linearized otimality system, using a Carleman inequality for degenerate non-autonomous equation obtanied by the authors previously.

Nash-Stackelberg controllability for coupled systems of degenerate equations in non-cylindrical domains

TL;DR

The article addresses local null controllability for a coupled, degenerate semilinear parabolic system in a time-dependent, non-cylindrical domain under a Stackelberg–Nash hierarchy with one leader and two followers. It first transforms the moving-domain problem to a cylindrical one, derives a Nash quasi-equilibrium via the linear–quadratic optimality system, and proves a Carleman inequality for the linearized adjoint system to obtain a global observability estimate. These linear results feed into Liusternik's inverse function theorem to establish local null controllability of the nonlinear system in a small-data regime, with a leader and follower controls driving both states to zero at the terminal time. The work extends hierarchical control methods to degenerate, coupled parabolic equations in moving domains, using a degenerate Carleman framework and a nonlinear inversion argument to achieve controllability.

Abstract

In this paper we investigate the Hierarchical null controllability of a coupled degenerate semilinear parabolic equation in domains which are moving in time. We show the local null controllability of the semilinear system using Liusternik's inverse function theorem. Nevertheless, the main difficulty is to adapt a Carleman estimate for the controllability of the linearized otimality system, using a Carleman inequality for degenerate non-autonomous equation obtanied by the authors previously.
Paper Structure (10 sections, 16 theorems, 259 equations)

This paper contains 10 sections, 16 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Characterization of Nash Equilibrium
  3. Passing to the cylindrical domain
  4. Characterization of the Nash-quase equilibrium and the optimality system
  5. Carleman Estimates
  6. Linearized system
  7. Global null controllability of the linearized system
  8. Local null controllability of the nonlinear system
  9. Proof of Proposition \ref{['prop:carleman_sistema']}.
  10. Proof of Proposition \ref{['propocarleman23']}.

Key Result

Theorem 1

Let us assume the hypothesis considered in the system Eq. (sistemanocilin4) and that, and Then, for any $T>0$ there exist $\varepsilon>0$ and a positive function $\rho_2(t)$ blowing up at $t=T$ such that, if $\rho_2 u^i_{d} \in L^2(\hat{\mathcal{O}}_{d} \times (0,T))$ and $u^0_1, u^0_2 \in H_a^1(\Omega)$ verify then there exists a control $\hat{h} \in L^2(\hat{\mathcal{O}}\times(0,T))$ and an

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Theorem 3
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • ...and 16 more