Optimal Boundary Control for Equations of Nonlinear

Acoustics

Christian Clason∗ Barbara Kaltenbacher† Irena Lasiecka‡

Slobodan Veljovic§

Abstract

Motivated by a medical application from lithotripsy, we study an optimal boundarycontrol problem given by Westervelt equation

− 1

c2D2

tu+ ∆u+b

c2∆(Dtu) = − βa

ρc4D2

tu2 in (0, T )× Ω (1)

modeling the nonlinear evolution of the acoustic pressure u in a smooth, bounded domainΩ ⊂ Rd, d ∈ 1, 2, 3. Here c > 0 is the speed of sound, b > 0 the diffusivity ofsound, ρ > 0 the mass density and βa > 1 the parameter of nonlinearity. We studythe optimization problem for existence of an optimal control and derive the first-ordernecessary optimality conditions.

In addition, all results are extended for the more general Kuznetsov equation

D2tψ − c2∆ψ = Dt

„b∆ψ +

1

c2B

2A(Dtψ)2 + |∇ψ|2

«(2)

given in terms of the acoustic velocity potential ψ.

Key words. optimal control, existence, nonlinear wave equation

∗Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz,Austria, (christian.clason@uni-graz.at)†Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz,

Austria, (barbara.kaltenbacher@uni-graz.at).‡Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA,

(il2v@virginia.edu).§Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz,

Austria, Fax: +43 316 380-9815, (slobodan.veljovic@uni-graz.at).

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978-1-4244-7827-9/10/$26.00 ©2010 IEEE 143