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

TL;DR

This paper develops a mathematical framework for controlling nonlinear ultrasonic wave propagation modeled by the Westervelt equation with fractional damping, enabling precise pressure field tracking in medical applications.

Contribution

It extends well-posedness theory to include Neumann boundary controls for the fractional Westervelt equation and derives optimality conditions for control problems.

Findings

Existence of globally optimal controls established.

Stability analysis with respect to target and regularization.

Derived first-order optimality conditions for control.

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.