We consider the optimal arrangement of two diffusion materials in a bounded open set \Omega \subset \BbbR N in order to maximize the energy. The diffusion problem is modeled by the p-Laplacian operator. It is well known that this type of problem has no solution in general and then that it is necessary to work with a relaxed formulation. In the present paper, we obtain such relaxed formulation using the homogenization theory; i.e., we replace both materials by microscopic mixtures of them. Then we get some uniqueness results and a system of optimality conditions. As a consequence, we prove some regularity properties for the optimal solutions of the relaxed problem. Namely, we show that the flux is in the Sobolev space H 1 (\Omega ) N and that the optimal proportion of the materials is derivable in the orthogonal direction to the flux. This will imply that the unrelaxed problem has no solution in general. Our results extend those obtained by the first author for the Laplace operator.

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For a diffusion problem modeled by the p-Laplacian operator, we are interested in obtaining numerically the two-phase material which maximizes the internal energy. We assume that the amount of the best material is limited. In the framework of a relaxed formulation, we present two algorithms, a feasible directions method and an alternating minimization method. We show the convergence for both of them, and we provide an estimate for the error. Since for p > 2 both methods are only well-defined for a finite-dimensional approximation, we also study the difference between solving the finite-dimensional and the infinite-dimensional problems. Although the error bounds for both methods are similar, numerical experiments show that the alternating minimization method works better than the feasible directions one.

We consider the optimal arrangement of two diffusion materials in a bounded open set Ω ⊂ R N in order to maximize the energy. The diffusion problem is modeled by the p-Laplacian operator. It is well known that this type of problems has no solution in general and then that it is necessary to work with a relaxed formulation. In the present paper we obtain such relaxed formulation using the homogenization theory, i.e. we replace both materials by microscopic mixtures of them. Then we get some uniqueness results and a system of optimality conditions. As a consequence we prove some regularity properties for the optimal solutions of the relaxed problem. Namely, we show that the flux is in the Sobolev space H 1 (Ω) N and that the optimal proportion of the materials is derivable in the orthogonal direction to the flux. This will imply that the unrelaxed problem has no solution in general. Our results extend those obtained by the first author for the Laplace operator.

A learning technique for finite horizon optimal control problems and its approximation based on polynomials is analyzed. It allows to circumvent, in part, the curse dimensionality which is involved when the feedback law is constructed by using the Hamilton-Jacobi-Bellman (HJB) equation. The convergence of the method is analyzed, while paying special attention to avoid the use of a global Lipschitz condition on the nonlinearity which describes the control system. The practicality and efficiency of the method is illustrated by several examples. For two of them a direct approach based on the HJB equation would be unfeasible.