MULTISCALE SINGULAR PERTURBATIONS AND HOMOGENIZATION OF OPTIMAL CONTROL PROBLEMS
Olivier Alvarez, Martino Bardi, Claudio Marchi · Series on advances in mathematics for applied sciences · 2008
The paper is devoted to singular perturbation problems with a finite number of scales where both the dynamics and the costs may oscillate. Under some coercivity assumptions on the Hamiltonian, we prove that the value functions converge locally uniformly to the solution of an effective Cauchy problem for a limit Hamilton-Jacobi equation and that the effective operators preserve several properties of the starting ones; under some additional hypotheses, their explicit formulas are exhibited. In some special cases we also describe the effective dynamics and costs of the limiting control problem. An important application is the homogenization of Hamilton-Jacobi equations with a finite number of scales and a coercive Hamiltonian. Keywords: Singular perturbations, viscosity solutions, Hamilton–Jacobi equations, dimension reduction, iterated homogenization, control systems in oscillating media, multiscale problems, oscillating costs.