An optimal control approach for ultrasound induced heating

The absorbed energy of ultrasound and microwaves is changed into heat in a dissipative medium. This wave field induced heating has numerous applications in industry and medicine. In these applications the ability to control the resulting heat distribution is essential. In this paper we design a controller for quadratic cost criteria in the ultrasound induced heating problem in inhomogeneous media. In this approach the time-harmonic pressure fields are computed from individual wave sources by solving the Helmholtz equation with the ultra-weak variational formulation. The equations that define the controller are derived from the Hamiltonian form of the system, which results in a large dimension non-linear control problem. As an example we study a numerical simulation that resembles an ultrasound therapy problem in 2D. The simulation shows that feasible solutions can be obtained with this approach.