SOME REMARKS ON STACKELBERG’S OPTIMIZATION

Abstract

We consider a distributed system governed by a parabolic equation, with a control v distributed over a subdomain [Formula: see text]. We assume that there are two goals we would like to achieve. The main one being of the “controllability” type, the “second one” expressing that the state of the system does not move “too far” from a given state. We introduce the following method. We divide v into two parts, say v1, v2, corresponding to a division of [Formula: see text] into two parts [Formula: see text]. Then we use the notion of Stackelberg’s optimization (introduced and used in Economy), where v2 is the follower and v1 the leader. Assuming that v1 is given, we optimize the second criterion with respect to v2, which gives v2=ℱ(v1) a function of v1. Then the first criterion becomes a function of v1 alone. It is of the approximate controllability type. In order to solve it, one has to prove a density theorem, which amounts to a uniqueness theorem, where we use a uniqueness theorem of Mizohata and regularity theorems of Cavalluci and of Matsuzawa.