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An optimal control problem for Maxwell's equations

Francesca Bucci, Matthias Eller

Abstract

This article is concerned with the optimal boundary control of the Maxwell system. We consider a Bolza problem, where the quadratic functional to be minimized penalizes the electromagnetic field at a given final time. Since the state is weighted in the energy space topology -- a physically realistic choice --, the property that the optimal cost operator does satisfy the Riccati equation (RE) corresponding to the optimization problem is missed, just like in the case of other significant hyperbolic partial differential equations; however, we prove that this Riccati operator as well as the optimal solution can be recovered by means of approximating problems for which the optimal synthesis holds via proper differential Riccati equations. In the case of zero conductivity, an explicit representation of the optimal pair is valid which does not demand the well-posedness of the RE, instead.

An optimal control problem for Maxwell's equations

Abstract

This article is concerned with the optimal boundary control of the Maxwell system. We consider a Bolza problem, where the quadratic functional to be minimized penalizes the electromagnetic field at a given final time. Since the state is weighted in the energy space topology -- a physically realistic choice --, the property that the optimal cost operator does satisfy the Riccati equation (RE) corresponding to the optimization problem is missed, just like in the case of other significant hyperbolic partial differential equations; however, we prove that this Riccati operator as well as the optimal solution can be recovered by means of approximating problems for which the optimal synthesis holds via proper differential Riccati equations. In the case of zero conductivity, an explicit representation of the optimal pair is valid which does not demand the well-posedness of the RE, instead.

Paper Structure

This paper contains 16 sections, 11 theorems, 109 equations.

Table of Contents

  1. Introduction and main result
  2. Main results
  3. An outline of the paper
  4. On well-posedness of Maxwell's equations
  5. A major well-posedness result. Weak solutions
  6. The operator underlying the free dynamics. Mild and strict solutions
  7. The input-to-state map, the control operator. Direct inequality
  8. The Green map
  9. The control operator $B$
  10. The optimal control problem: The case $U=H^{1/2}_\mathrm{tan}(\Gamma)$
  11. The open-loop optimal solution
  12. Closed-loop form of the optimal control
  13. The optimal cost operator, feedback formula
  14. Approximating $P(t)$ via solutions to proper Riccati equations
  15. Problem $\mathbf{\mathcal{P}}_n$
  16. ...and 1 more sections

Key Result

Theorem 1.2

With reference to Problem p:problem_s, the following statements hold true for any $s\in [0,T)$.

Theorems & Definitions (21)

  • Theorem 1.2: Main result
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 11 more