Appl Math Optim 46:263–290 (2002)
2002 Springer-Verlag New York Inc.
A Posteriori Error Estimates in Time-Domain Decomposition of
Final Value Optimal Control of the Acoustic Wave Equation
J. E. Lagnese
and G. Leugering
Department of Mathematics, Georgetown University,
Washington, DC 20057, USA
Fachbereich Mathematik, Technische Universit¨at Darmstadt,
Schlossgartenstrasse 7, D-64289 Darmstadt, Germany
Abstract. We consider a boundary optimal control problem for the acoustic wave
equation with a ﬁnal value cost criterion. A time-domain decomposition procedure
for the corresponding optimality system is introduced, which leads to a sequence
of uncoupled optimality systems of local-in-time optimal control problems. The
process is inherently parallel and is suitable for real-time control applications. Con-
vergence of the local solutions and controls to the global ones was established in an
earlier publication. In this paper, a posteriori error estimates of the difference be-
tween the local solutions and the global one are obtained in terms of the mismatch
of the nth iterates, or of successive iterates, across the time-domain break points.
Key Words. Acoustic wave equation, Optimal control, Domain decomposition,
A posteriori error estimates.
AMS Classiﬁcation. 65N55, 49M27, 35Q60.
We consider optimal boundary control problems for wave equations in multidimensional
spaces. Problems of this type frequently occur, e.g., in structural acoustics or ﬂexible
structures, and they have been studied extensively in the literature over the last 30 years.
The research of J.E.L. was supported by the National Science Foundation through Grant DMS-9972034.
The research of G.L. was supported by DFG Grants Le595/12-2 and Le595/13-1.