## F. Maestre - A. Münch - P. Pedregal

# A spatio-temporal design problem for a damped wave equation

created by maestre on 30 Apr 2008

modified on 03 May 2008

[

BibTeX]

*Accepted Paper*

**Inserted:** 30 apr 2008

**Last Updated:** 3 may 2008

**Journal:** SIAP

**Volume:** 68

**Number:** 1

**Pages:** 109-132

**Year:** 2007

**Abstract:**

We analyze in this work a spatio-temporal optimal design problem governed by a
linear damped one-dimensional wave equation. The problem consists of simultaneously seeking the
spatio-temporal layout of two isotropic materials and the static position of the damping set in order
to minimize a functional depending quadratically on the gradient of the state. The lack of classical
solutions for this kind of nonlinear problem is well known. We examine a well-posed relaxation
by using the representation of a two-dimensional divergence-free vector as a rotated gradient. We
transform the original optimal design problem into a nonconvex vector variational problem. By means
of gradient Young measures we compute an explicit form of the constrained quasi convexification of
the cost density. Moreover, this quasi convexification is recovered by first order laminates which give
the optimal distribution of materials and damping set at every point. Finally, we analyze the relaxed
problem, and some numerical experiments are performed. The novelty here lies in the optimization
with respect to two independent subdomains, and our contribution consists of understanding their
mutual interaction.

**Keywords:**
relaxation, wave equation, optimal design, Young measure

**Download:**