Optimal boundary control for equations of nonlinear acoustics

Motivated by a medical application from lithotripsy, we study an optimal boundary control problem given by Westervelt equation − ¹/c²D²t u + Δu + b/c²Δ(Dt u) = − βa/ρc⁴D²t u² in (0, T) × Ω (1) modeling the nonlinear evolution of the acoustic pressure u in a smooth, bounded domain Ω ⊂ R^d, d ∊ {1, 2, 3}. Here c > 0 is the speed of sound, b > 0 the diffusivity of sound, ρ > 0 the mass density and βa > 1 the parameter of nonlinearity. We study the optimization problem for existence of an optimal control and derive the first-order necessary optimality conditions. In addition, all results are extended for the more general Kuznetsov equation D²tψ − c²Δψ = Dt(bΔψ + 1/c² B/2A (Dtψ)² + |∇ψ|²) (2) given in terms of the acoustic velocity potential ψ.