Homogenization results for a deterministic multi-domains periodic control problem

Abstract

We consider homogenization problems in the framework of deterministic optimal control when the dynamics and running costs are completely different in two (or more) complementary domains of the space R N . For such optimal control problems, the three first authors have shown that several value functions can be defined, depending, in particular, of the choice is to use only "regular strategies" or to use also "singular strategies". We study the homogenization problem in these two different cases. It is worth pointing out that, if the second one can be handled by usual partial differential equations method "à la Lions-Papanicolaou-Varadhan" with suitable adaptations, the first case has to be treated by control methods (dynamic programming).

A Bellman Approach for Regional Optimal Control Problems in $\mathbb{R}^N$

“…Compared to [6], we have modified the strategy of the comparison proofs by emphasizing the role of a "local comparison result" which is given in the Appendix. There are several reasons to do so : such local results are useful for applications, for example in homogenization problems which we consider in a forthcoming work with N. Tchou [7]; in such applications the use of the perturbed test-function of L. C. Evans [21,22] requires (or is far more simpler with) such local comparison results. On the other hand we have to handle, at the same time, a more complex geometry than in [6] and a weaker controlability assumption (which implies that the sub solutions are not automatically Lipschitz continuous) and to argue locally allow to flatten the interface and use a double regularization procedure on the subsolutions in the tangent variables, first by supconvolution to reduce to the Lipschitz continuous case and then by usual mollification.…”

Homogenization of a transmission problem with Hamilton–Jacobi equations and a two-scale interface. Effective transmission conditions

“…the admissible dynamics are the ones corresponding to the subdomain Ω L ε,ε and entering Ω L ε,ε , or corresponding to the subdomain Ω R ε,ε and entering Ω R ε,ε . Hence the situation differs from those studied in the articles of G. Barles, A. Briani and E. Chasseigne [8,9] and of G. Barles, A. Briani, E. Chasseigne and N. Tchou [11], in which mixing is allowed at the interface. The optimal control problem under consideration has been first studied in [25]: the value function is characterized as the viscosity solution of a Hamilton-Jacobi equation with special transmission conditions on Γ ε,ε ; a comparison principle for this problem is proved in [25] with arguments from the theory of optimal control similar to those introduced in [8,9].…”

Effective transmission conditions for Hamilton–Jacobi equations defined on two domains separated by an oscillatory interface

“…the admissible dynamics are the ones corresponding to the subdomain Ω L ε and entering Ω L ε , or corresponding to the subdomain Ω R ε and entering Ω R ε . Hence the situation differs from those studied in the articles of G. Barles, A. Briani and E. Chasseigne [5,6] and of G. Barles, A. Briani, E. Chasseigne and N. Tchou [7], in which mixing is allowed at the interface. The optimal control problem under consideration has been first studied in [16]: the value function is characterized as the viscosity solution of a Hamilton-Jacobi equation with special transmission conditions on Γ ε ; a comparison principle for this problem is proved in [16] with arguments from the theory of optimal control similar to those introduced in [5,6].…”

“…Choose m(δ) = Π R (z 2 , p 2 ) − q δ ≥ 0 and consider the function w R : w R (y) = q δ (y 1 + g(y 2 )), which is of class C 2 . From the choice of q δ , for any y ∈ R 2 , 7) and for any y ∈ {ȳ 1 } × R,…”