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Optimal control of a parabolic equation with a nonlocal nonlinearity

Cyrille Kenne, Landry Djomegne, Gisèle Mophou

Abstract

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established.

Optimal control of a parabolic equation with a nonlocal nonlinearity

Abstract

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established.
Paper Structure (12 sections, 20 theorems, 152 equations)

This paper contains 12 sections, 20 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Existence results
  4. Resolution of the optimization problem
  5. Existence of optimal solutions
  6. Optimality conditions
  7. Regularity results for the control-to-state operator
  8. The adjoint equation
  9. First order necessary optimality conditions
  10. Second order optimality conditions
  11. Second order necessary optimality conditions
  12. Concluding remarks

Key Result

Lemma 2.1

Let $y\in W(0,T)$. Then, $y\in L^{\sigma}(Q)$, with $\sigma=\frac{2(n+2)}{n}$. In addition, there exists a constant $C=C(n)$ such that the following estimate holds.

Theorems & Definitions (44)

  • Remark 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 34 more