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Optimal Neumann boundary and distributed control of the Westervelt equation with time-fractional attenuation

Vanja Nikolić, Belkacem Said-Houari

TL;DR

The paper addresses optimal control of nonlinear acoustic waves modeled by the Westervelt equation with time-fractional attenuation, encompassing both Neumann boundary and distributed controls. It develops a rigorous forward analysis, including two regularity regimes and a fixed-point argument to handle the quasilinear, fractionally damped state, and it extends well-posedness to inhomogeneous Neumann data via a data-extension technique. Existence of globally optimal controls is proven, with stability results under perturbations of the target pressure and vanishing regularization, and adjoint-based first-order optimality conditions are derived, despite the state-dependent adjoint coefficients. The results provide a solid theoretical foundation for adjoint-based optimization in fractionally damped Westervelt models, with potential guidance for numerical algorithms and applications in medical ultrasound where precise energy deposition is essential.

Abstract

Optimal control of nonlinear acoustic waves is relevant in many medical ultrasound technologies, ranging from cancer therapy to targeted drug delivery, where it can help guide the precise deposition of acoustic energy. In this work, we study Neumann boundary and distributed control problems for tracking a prescribed pressure field governed by the Westervelt equation with time-fractional dissipation. This model captures nonlinear ultrasonic wave propagation in biological media and accounts for the experimentally observed power-law attenuation. We begin by extending the existing well-posedness theory for time-fractional equations to include inhomogeneous Neumann boundary data used as control inputs, which requires constructing an appropriate data extension and regularization. Using these analytical results for the forward problem, we prove the existence of globally optimal controls and analyze the stability of the optimization problem with respect to perturbations in the target pressure field and to vanishing regularization parameters. Finally, we investigate the associated adjoint equation, which has state-dependent coefficients, and use it to derive first-order necessary optimality conditions.

Optimal Neumann boundary and distributed control of the Westervelt equation with time-fractional attenuation

TL;DR

The paper addresses optimal control of nonlinear acoustic waves modeled by the Westervelt equation with time-fractional attenuation, encompassing both Neumann boundary and distributed controls. It develops a rigorous forward analysis, including two regularity regimes and a fixed-point argument to handle the quasilinear, fractionally damped state, and it extends well-posedness to inhomogeneous Neumann data via a data-extension technique. Existence of globally optimal controls is proven, with stability results under perturbations of the target pressure and vanishing regularization, and adjoint-based first-order optimality conditions are derived, despite the state-dependent adjoint coefficients. The results provide a solid theoretical foundation for adjoint-based optimization in fractionally damped Westervelt models, with potential guidance for numerical algorithms and applications in medical ultrasound where precise energy deposition is essential.

Abstract

Optimal control of nonlinear acoustic waves is relevant in many medical ultrasound technologies, ranging from cancer therapy to targeted drug delivery, where it can help guide the precise deposition of acoustic energy. In this work, we study Neumann boundary and distributed control problems for tracking a prescribed pressure field governed by the Westervelt equation with time-fractional dissipation. This model captures nonlinear ultrasonic wave propagation in biological media and accounts for the experimentally observed power-law attenuation. We begin by extending the existing well-posedness theory for time-fractional equations to include inhomogeneous Neumann boundary data used as control inputs, which requires constructing an appropriate data extension and regularization. Using these analytical results for the forward problem, we prove the existence of globally optimal controls and analyze the stability of the optimization problem with respect to perturbations in the target pressure field and to vanishing regularization parameters. Finally, we investigate the associated adjoint equation, which has state-dependent coefficients, and use it to derive first-order necessary optimality conditions.

Paper Structure

This paper contains 18 sections, 13 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Theoretical preliminaries in fractional calculus
  3. Analysis of wave equations with time-fractional attenuation
  4. Analysis of a linearized wave problem: Lower-regularity regime
  5. Analysis of a linearized wave problem: Higher-regularity regime
  6. Well-posedness of the state problem
  7. Existence of optimal controls
  8. Stability with respect to perturbations
  9. First-order necessary optimality conditions
  10. The adjoint problem
  11. Differentiability of the control-to-state mapping
  12. Derivative of the reduced objective
  13. Optimality conditions for the Numann boundary control problem
  14. Optimality conditions for the distributed control problem
  15. Details on the derivation of energy estimates
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let $T>0$ and let $\Omega \subset \mathbb{R}^d$ with $d \in \{1,2,3\}$ be a Lipschitz-regular bounded domain. Let $c>0$, $b \in (0, \bar{b}]$ for some $\bar{b}>0$, and $\alpha \in (0,1)$. Assume that the variable coefficients in linearized problem satisfy and that the non-degeneracy condition on $\mathfrak{a}$ in nondegeneracy assumption frakm frakm holds. Furthermore, let $F \in L^2(0,T; L^2(\Om

Theorems & Definitions (25)

  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: Compatible regularized boundary data
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.1
  • Remark 1: On the smallness assumption
  • ...and 15 more