  Classical models for the propagation of ultrasound waves are the Westervelt
equation, the Kuznetsov and the Khokhlov-Zabolotskaya-Kuznetsov equations. The
Jordan-Moore-Gibson-Thompson equation is a prominent example of a Partial
Differential Equation (PDE) model which describes the acoustic velocity
potential in ultrasound wave propagation, 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. Aiming at the understanding of the fully
quasilinear PDE, a great deal of attention has been recently devoted to its
linearization -- referred to in the literature as the Moore-Gibson-Thompson
equation -- whose mathematical analysis is also of independent interest, posing
already several questions and 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 $L^2$ in time; this leads to a singular control problem and to
non-standard Riccati equations. In spite of the unfavourable combination of the
semigroup describing the free dynamics that is not analytic, with the
challenging pattern displayed by the dynamics subject to boundary control, a
feedback synthesis of the optimal control as well as well-posedness of operator
Riccati equations are established.
