SOME REMARKS ON STACKELBERGS OPTIMIZATION

We consider a distributed system governed by a parabolic equation, with a control v distributed over a subdomain O. 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 v(1), v(2), corresponding to a division of O into two parts O-1, O-2. Then we use the notion of Stackelberg's optimisation (introduced and used in Economy), where v(2) is the follower and v(1) the leader. Assuming that v(1) is given, we optimize the second criterion with respect to v(2), which gives v(2) = F(v(1)) a function of v(1). Then the first criterion becomes a function of v(1) 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.