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J.-P.Lohéac

An introduction to the controllability of partial differential equations

The goal of this course is to teach some basic methodologies for analysing problems of control of partial differential equations.

Control problems arise in many different contexts and ways. Roughly speaking the controllability problem consists in analysing whether the solution can be driven from some initial data to a given final target by means of a suitable control acting on data of the problem (i.e. the right-hand side or the boundary conditions).

When dealing with controllability problems, one has to distinguish between finite-dimensional systems modelled by ordinary differential equations and infinite-dimensional distributed systems described by means of partial differential equations.

Most of this course deal with problems related with partial differential equations. However, we will start by presenting some of the basic problems and tools of control theory for finite-dimensional systems. Especially we shall develop a variational approach so that the control can be built by minimizing a suitable quadratic functional. The main difficulty is to show that this functional is coercive. This leads to the so-called observability property of the adjoint problem.

After that, we shall extend these results to control of partial differential equations. We mainly introduce some variants as approximate, exact and null controllability and observability and unique continuation property.

In addition we shall also discuss some numerical aspects of the problem.

Bibliography

Komornik, V., 1994, Exact controllability and stabilization; the multiplier method. Masson-John Wiley, Paris.

Lions, J.-L., 1986, Contrôlabilité exacte, perturbation et stabilisation de systèmes distribués. Masson, Paris.