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Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms

Juan Limaco, Rafael Martins Lobosco, Luis P. Yapu

TL;DR

The paper addresses local null controllability for semilinear, nonlocally coupled parabolic PDEs driven by a single internal control. It combines Carleman-based observability estimates for the linearized, nonconstant-coefficient system with a Kakutani fixed-point argument to handle nonlinear terms, establishing a fixed-point control that steers the state to zero for small initial data. Key contributions include extending controllability results to variable coefficients and nonlocal kernels, deriving explicit control bounds, and discussing boundary-controllability within the same nonlocal framework. These results have potential implications for finance-inspired PDE models with regime-switching kernels and nonlocal interactions, and they lay groundwork for further generalizations to larger networks and fewer controls.

Abstract

This paper extends our previous controllability results for a class of coupled linear parabolic systems with nonlocal interactions, motivated by applications in finance such as generalized Black--Scholes models. We establish local null controllability at a fixed time T>0 for a class of semilinear, nonlocally coupled systems driven by a single internal control acting on one component. The proof combines Kakutani's fixed-point theorem with a controllability/observability estimate for the associated linearized dynamics. In addition, we obtain controllability for a broader class of linear systems than those considered in the first article. The paper concludes with remarks on boundary controllability within the same nonlocal framework and with perspectives for future research.

Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms

TL;DR

The paper addresses local null controllability for semilinear, nonlocally coupled parabolic PDEs driven by a single internal control. It combines Carleman-based observability estimates for the linearized, nonconstant-coefficient system with a Kakutani fixed-point argument to handle nonlinear terms, establishing a fixed-point control that steers the state to zero for small initial data. Key contributions include extending controllability results to variable coefficients and nonlocal kernels, deriving explicit control bounds, and discussing boundary-controllability within the same nonlocal framework. These results have potential implications for finance-inspired PDE models with regime-switching kernels and nonlocal interactions, and they lay groundwork for further generalizations to larger networks and fewer controls.

Abstract

This paper extends our previous controllability results for a class of coupled linear parabolic systems with nonlocal interactions, motivated by applications in finance such as generalized Black--Scholes models. We establish local null controllability at a fixed time T>0 for a class of semilinear, nonlocally coupled systems driven by a single internal control acting on one component. The proof combines Kakutani's fixed-point theorem with a controllability/observability estimate for the associated linearized dynamics. In addition, we obtain controllability for a broader class of linear systems than those considered in the first article. The paper concludes with remarks on boundary controllability within the same nonlocal framework and with perspectives for future research.

Paper Structure

This paper contains 4 sections, 10 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Analysis of the Controllability of the Linearized System of (1)
  3. Proof of Theorem \ref{['maintheorem0']}
  4. Related problems and additional comments

Key Result

Theorem 1

Let $T>0$. Suppose that hypothesis H_0 and H2 are satisfied. Then there exists $\delta = \delta(T)>0$ such that, for any initial data $y_0, z_0 \in L^2(\Omega)$ satisfying and for any $F(r,s), G(r,s) : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, $C^1$-functions fulfilling for all $(r,s) \in \mathbb{R}^2$, with $\frac{\partial G}{\partial r} (r,s) \neq 0$ in sistema0, $F(0,0)=0$, $G(0,0)=0$, th

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Proposition 2: GlobalCarleman
  • proof
  • Proposition 3: biccari2019
  • Proposition 4
  • proof
  • Remark 1
  • ...and 7 more