# Control and optimal design

## On the optimal distribution of null controls for the heat and wave equations

We review here the problem which consists to minimize the norm of HUM controls for the wave and heat-like equations with respect to their support. The problem reduces to an optimal design problem for a dynamical equation defined on a bounded cylinder. As is well-known, this type of problem may have no solution in the class of characteristic functions. First, we show how, by the use of convex analysis, one may associate a well-posed reformulation in the class of density functions. Then, we present some numerical experiments that allow to separate qualitatively the heat situation from the wave one. We also discuss some recent contributions where the support evolves in time, the minimization over weighted Carleman type norms and the related stabilization issue.

## Relaxation and discretization of control problems in the coefficients with a nonlinear functional in the gradient

We consider a control problem for a linear elliptic PDE equation where the control variable is the corresponding diffusion matrix. It is well known that this type of problems has not solution in general and that for a sequentially continuous functional in the weak topology of the Sobolev space $H_0^1$, a relaxation can be obtained via homogenization by replacing the original set of controls by its $H$-closure. When this continuity assumption of the functional is not satisfied, we show that this relaxation can also be carried on but now it is also necessary to introduce a certain extension of the original functional to the $H$-closure of the set of original controls. This extension is not explicitly known in general, which complicates the numerical resolution of the problem. We give some strategies permitting to overcome this difficulty.

In this talk I'll review the lasted developments in the application of the control theory to the optimal shape design of airplanes. There is a well consolidated theory about how to apply the continuous and discrete adjoint methodology to optimal shape design, however the application of these techniques to real industrial problems is in an initial stage due to the complexity of the entire optimization loop (numerical grids which exceed 10 millions cells, Continuous Adjoint and Automatic Differentiation bottlenecks, the necessity of efficient design variables in a 3D environment or more robust parallel algorithms). To sum up, in my talk I'll do a review of the current optimal shape design technology for systems governed by the Euler equations and Reynolds-averaged NavierStokes equations (RANS), and also I'll highlight the areas with a special relevance, from the application side, for future theoretical studies.