Some control problems in electromagnetics and fluid dynamics.

In recent years, as a consequence of the dramatic increases in computing power and of the continuing refinement of the numerical algorithms available, it has become possible the numerical treatment of control problems for systems governed by partial differential equations see, for example, [1], [3], [4], [5], [8]. The importance of these mathematical problems in many applications in science and technology cannot be overemphasized. The most common approach to a control problem for a system governed by partial differential equations is to see the problem as a constrained nonlinear optimization problem in infinite dimension. After discretization the problem becomes a finite dimensional constrained nonlinear optimization problem that can be attacked with the usual iterative methods of nonlinear optimization, such as Newton or quasi-Newton methods. Note that the problem of the convergence, when the ``discretization step goes to zero", of the solutions computed in finite dimension to the solution of the infinite dimensional problem is a separate question and must be solved separately. When this approach is used an objective function evaluation in the nonlinear optimization procedure involves the solution of the partial differential equations that govern the system. Moreover the evaluation of the gradient or Hessian of the objective function involves the solution of some kind of sensitivity equations for the partial differential equations considered. That is the nonlinear optimization procedure that usually involves function, gradient and Hessian evaluation is computationally very expensive. This fact is a serious limitation to the use of control problems for systems governed by partial differential equations in real situations. However the approach previously described is very straightforward and does not use the special features present in each system governed by partial differential equations. So that, at least in some special cases, should be possible to improve on it. The purpose of this paper is to point out a problem, see [6], [2], where a new approach, that greatly improves on the previously described one, has been introduced and to suggest some other problems where, hopefully, similar improvements can be obtained. In section 1.2 we summarize the results obtained in [6], [2] and in section 1.3 we present two problems that we belive can be approached in a way similar to the one described in [6], [2].