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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.