Winter School on Transport Equations
and Control Theory for PDEs


An Introduction to Control Theory
for Hyperbolic-like PDEs

These lectures will introduce the audience to the problems of a-priori, inverse-type inequalities, such as they arise in the Control Theory of Partial Differential Equations for hyperbolic and Petrowski-type PDEs, defined on a multi-dimensional bounded domain.

These include:
(i) continuous observability inequalities;
(ii) stabilization inequalities. As a consequence, one obtains exact controllability results with control acting on a portion of the boundary; energy decay (stabilization) results with dissipation active on a portion of the boundary; and, a-fortiori, global uniqueness theorems for over-determined problems.

Linear and semi-linear PDEs to be considered include: second order hyperbolic equations, first order hyperbolic systems; Schrodinger equations; various plate-like equations with finite or infinite speed of propagation; linear and semi-linear models; etc, depending on time-constraints.

More specific topics will include:

(a) Carleman-type inequalities and from here a-priori continuous observability/ stabilization inequalities for linear equations: from canonical models to general models with variable coefficients and energy-level terms. The linear estimates are derived under essentially minimal assumptions imposed on the regularity of the coefficients, an within an essentially self-contained approach. This is a critical starting point for the non-linear analysis of the related control problems, to be described in (b) and (c) below.

(b) Exact controllability for semi-linear waves, plates equations.

(c) Uniform stability and energy decay estimates for hyperbolic and Petrowski problems (waves, plates, and even shells if time permits..)

References