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 $L^2$ in time; this leads to a singular control problem and to non-standard Riccati equations.

KEYWORDS:

• Ultrasound waves
• singular boundary control
• non-standard Riccati equations

Acknowledgements

The authors are grateful to Barbara Kaltenbacher, whose work has provided motivation for studying control problems associated with the SMGT acoustic model. Inspiring and illuminating mathematical conversations of both authors with Barbara are gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of F.B. was partially supported by the Università degli Studi di Firenze under the Project Analisi e controllo di sistemi di Equazioni a Derivate Parziali di evoluzione, and by the Groupement de Recherche Européen (GDRE) ConEDP (Control of PDEs). F.B. is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), whose occasional support is acknowledged. The research of I.L. was partially supported by the NSF-DMS 1713506.