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First and second-order optimality conditions for a bilinear controlled wave equation on an infinite horizon

Redouane El Mezegueldy, Zakarya Dardour

Abstract

This paper investigates the optimal control of a bilinear damped wave equation over an infinite time horizon. We establish the well-posedness of the controlled system and derive uniform energy estimates. The existence of optimal controls is proven by constructing a minimizing sequence. We prove that the control-to-state mapping is twice continuously Fréchet differentiable, which enables the derivation of first-order necessary optimality conditions in the form of a variational inequality and a pointwise projection formula. Furthermore, we establish second-order necessary and sufficient conditions: the nonnegativity of the Hessian of the cost functional is shown to be a necessary condition for local optimality, while the coercivity of this Hessian constitutes a sufficient condition. These results provide a complete characterization of local optimality for bilinear hyperbolic control systems over infinite time horizons on bounded spatial domains.

First and second-order optimality conditions for a bilinear controlled wave equation on an infinite horizon

Abstract

This paper investigates the optimal control of a bilinear damped wave equation over an infinite time horizon. We establish the well-posedness of the controlled system and derive uniform energy estimates. The existence of optimal controls is proven by constructing a minimizing sequence. We prove that the control-to-state mapping is twice continuously Fréchet differentiable, which enables the derivation of first-order necessary optimality conditions in the form of a variational inequality and a pointwise projection formula. Furthermore, we establish second-order necessary and sufficient conditions: the nonnegativity of the Hessian of the cost functional is shown to be a necessary condition for local optimality, while the coercivity of this Hessian constitutes a sufficient condition. These results provide a complete characterization of local optimality for bilinear hyperbolic control systems over infinite time horizons on bounded spatial domains.
Paper Structure (11 sections, 13 theorems, 108 equations)

This paper contains 11 sections, 13 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Well-posedness and regularity of the state equation
  3. Differentiability analysis
  4. The adjoint equation
  5. Differentiability of the control-to-state mapping
  6. Differentiability of the cost functional
  7. Existence and first-order optimality conditions
  8. Existence of optimal controls
  9. First-order necessary conditions
  10. Second-order optimality conditions
  11. Conclusion

Key Result

Lemma 2.3

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then there exists a constant $c_{\Omega}>0$ depending only on $\Omega$ such that

Theorems & Definitions (17)

  • Definition 2.1: Weak solution
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.3
  • ...and 7 more