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Stackelberg-Nash strategy for the null controllability of semilinear degenerate equations in non-cylindrical domains

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

TL;DR

This work addresses local null controllability for a semilinear degenerate parabolic equation defined on a moving (non-cylindrical) domain, where the diffusion $a(x)$ vanishes at the boundary and behaves like $a(x)\sim x^{\alpha}$ with $\alpha\in(0,1)$. It adopts a Stackelberg-Nash hierarchy with a leader $h$ and two followers $v^1,v^2$, transforming the moving domain to a cylindrical one via a time-dependent diffeomorphism, and proving null controllability through Carleman estimates for the resulting non-autonomous degenerate system and Liusternik's inverse function theorem for the nonlinear problem. The authors establish Nash quasi-equilibria for the followers, derive the associated optimality system, and show that large follower penalties yield a true Nash equilibrium; they also prove global null controllability for the linearized system with explicit weighted estimates and then obtain local controllability for the nonlinear system. These results extend controllability theory to degenerate parabolic equations in moving domains and provide a rigorous framework for hierarchical control under non-autonomous, time-varying diffusion. The work has potential implications for diffusion processes with boundary degeneration in evolving spatial domains and for applications requiring multi-level control structures.

Abstract

In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a semilinear parabolic equation in one-dimension defined in a non-cylindrical domain where the diffusion coefficient degenerates at one point of the boundary. The linearized degenerated system is treated using a Carleman inequality for degenerated non-autonomous systems proved by the autors in [19] and the local controllability of the semilinear system is obtained using Liusterniks inverse function theorem.

Stackelberg-Nash strategy for the null controllability of semilinear degenerate equations in non-cylindrical domains

TL;DR

This work addresses local null controllability for a semilinear degenerate parabolic equation defined on a moving (non-cylindrical) domain, where the diffusion vanishes at the boundary and behaves like with . It adopts a Stackelberg-Nash hierarchy with a leader and two followers , transforming the moving domain to a cylindrical one via a time-dependent diffeomorphism, and proving null controllability through Carleman estimates for the resulting non-autonomous degenerate system and Liusternik's inverse function theorem for the nonlinear problem. The authors establish Nash quasi-equilibria for the followers, derive the associated optimality system, and show that large follower penalties yield a true Nash equilibrium; they also prove global null controllability for the linearized system with explicit weighted estimates and then obtain local controllability for the nonlinear system. These results extend controllability theory to degenerate parabolic equations in moving domains and provide a rigorous framework for hierarchical control under non-autonomous, time-varying diffusion. The work has potential implications for diffusion processes with boundary degeneration in evolving spatial domains and for applications requiring multi-level control structures.

Abstract

In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a semilinear parabolic equation in one-dimension defined in a non-cylindrical domain where the diffusion coefficient degenerates at one point of the boundary. The linearized degenerated system is treated using a Carleman inequality for degenerated non-autonomous systems proved by the autors in [19] and the local controllability of the semilinear system is obtained using Liusterniks inverse function theorem.

Paper Structure

This paper contains 11 sections, 15 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Change of variables
  3. CHARACTERIZATION OF NASH QUASI-EQUILIBRIUM AND EQUILIBRIUM
  4. Caracterization of the Nash equilibrium
  5. Nash quasi-equilibria and Nash equilibria
  6. Null controllability of the linearized system
  7. Hilbert spaces in the divergence case
  8. Carleman estimates
  9. Global null controllability of the linearized system
  10. Local null controllability of the nonlinear system
  11. Final remarks and some problems

Key Result

Theorem 1

Let us assume the hypothesis considered in the setting of Eq. (eq:PDE) and that, for $i=1,2$, and Then, for any $T>0$ there exist $\varepsilon>0$ and a positive function $\rho(t)$ blowing up at $t=T$ such that, if $\rho y_{i,d} \in L^2(O_d \times (0,T))$ and $u_0 \in H_a^1(\Omega_0)$ verifies then there exist a control $\hat{h} \in L^2(\hat{O}\times(0,T))$ and associated Nash quasi-equilibrium

Theorems & Definitions (22)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Corollary 1
  • proof
  • ...and 12 more