Résumé en anglais

This paper is concerned with optimal control problems for distributed-parameter systems over a bounded domain Ω. The state equations are linear partial differential equations of elliptic type, the controls and the cost are distributed over the domain or over its boundary. The controls are bounded and appear in the boundary conditions or in the right-hand member of the state equations.It is easily seen that such problems do usually not admit optimal controls. However, minimizing sequences of controls always exist. For brevity, let us call “optimal state” the state associated with an optimal control, and “minimizing sequence of states” the sequence of states corresponding to a minimizing sequence of controls. Using compacity arguments in Sobolev spaces, it is easily shown that every minimizing sequence of states has cluster points in. The purpose of this paper is to study those cluster points.It is shown that the set of cluster points of all the minimizing sequences of states of a given control problem is the set of optimal states of a new control problem of the same type, called the relaxed problem. This second problem is constructed directly from the first one, by a kind of point-by-point convexification, and it is also proved that both problems have the same value. The proofs rely heavily on convexity arguments. A sufficient condition for the first problem to admit optimal controls is deduced. Finally, two different ways of constructing the relaxed problem are given and are shown to be equivalent to the first one.