Feedback control of the acoustic pressure in ultrasonic wave propagation

ABSTRACTThe 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 literatur...