Long time behavior of a two-phase optimal design for the heat equation

We consider a two-phase isotropic optimal design problem within the context of the transient heat equation. The objective is to minimize the average of the dissipated thermal energy during a fixed time interval [0,T]. The time-independent material properties are taken as design variables. A full relaxation for this problem was established in [A. Münch, P. Pedregal, and F. Periago, J. Math. Pures Appl. (9), 89 (2008), pp. 225-247] by using the homogenization method. In this paper, we study the asymptotic behavior as T goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed de sign of the corresponding two-phase optimization problem for the stationary heat equation. Next we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the microstructure of the optimal designs, which appears in the form of a sequential laminate of rank at most N, the spatial dimension. An asymptotic analysis of the optimality conditions lets us prove that, for T large enough, the order of lamination is, in fact, of at most N - 1. Several numerical experiments in two dimensions complete our study.