Feedback control of the acoustic pressure in ultrasonic wave propagation

Abstract: The Jordan–Moore–Gibson–Thompson equation is a prominent example of a Partial Differential Equation model which describes the acoustic velocity potential in ultrasound wave propagation, and where the paradox of infinite speed of propagation of thermal signals is eliminated; the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law, accounts for its being of third order in time. A great deal of attention has been recently devoted to its linearization – referred to in the literature as the Moore–Gibson–Thompson equation – whose analysis poses already several questions and mathematical challenges. In this work, we consider and solve a quadratic control problem associated with the linear equation, formulated consistently with the goal of keeping the acoustic pressure close to a reference pressure during ultrasound excitation, as required in medical and industrial applications. While optimal control problems with smooth controls have been considered in the recent literature, we aim at relying on controls which are just square integrable in time; this leads to a singular control problem and to non-standard Riccati equations.