- George Avalos, University of Nebraska-Lincoln
- Francesca Bucci, Universita' di Firenze

"Novel directions in control of evolutionary PDE problems"

We intend that our minisymposia should deal with topics concerning the mathematical control of partial di¤erential equation (PDE) systems. In particular, the emphasis here will be on those PDEs which have been used to describe various phenomena in the physical sciences and engineering.

It is anticipated that our Speakers will broach in our Sessions ostensibly classical control theoretic

subjects, which however will be framed within a modern PDE setting, and rigorously developed

by state of the art time/frequency domain multiplier methods and pseudodifferential/PDE control techniques.

Indeed, the participants in our Minisymposia will be internationally recognized pioneers and contributors in the mathematical control of in nite dimensional systems.

In addition to the intrinsic merit of our proposed forum to have renowned experts in PDE control theory present their recent research, there is the possibility for further advancement in the field, by virtue of the

opportunity provided in our Minisymposia for discussion and future collaboration.

Examples of such problems which would fall under the scope of our proposed Sessions include

the following.

(i) The optimal control of PDEs with respect to given quadratic or non-quadratic cost functionals; with possible characterizations of these optimal controls being provided via appropriate solutions to the Riccati equation.

In particular, there would be an emphasis on PDE systems whose characteristics are of mixed type; in which case optimal control theories currently found in the literature would not avail in the associated PDE and optimzation analysis, inasmuch as existing PDE optimal control results are critically dependent upon the characteristics of the PDE dynamics involved.

Examples of such mixed-type PDE control problems include uid-structure PDE systems with boundary control effected in the so-called Polya tensor; and composite sandwichstructures which are composed of a multiplicity of elastic PDEs, some of which are under the influence of boundary control.

(ii) Stabilization of given PDE dynamics through the agency of dissipation-enhancing boundary feedback control mechanisms. Along with those uncontrolled PDE systems which exhibit a underlying conservation of energy, it is possible that the free dynamics which are to be feedback- controlled would actually manifest heat-generating (feedback) sources (or an anti-conservation, if you will). The control objective would then be to demonstrate that the incorporation of either boundary and/or strictly localized interior control, each in feedback form, will induce some bene ficient stabilizing effect upon the controlled PDE solutions, in long time. Such a stabilization could be realized as an outright uniform decay of the PDE solutions, as time goes to in nity; or possibly a global attracting set to which the solutions tend in the course of evolution. (The latter possibility would arise in the case of PDE systems under the inuence of non-Lipschitz and non-dissipative nonlinearities.)

(iii) The steering or controlling of solutions of certain PDE dynamics through the implementation of control functions, which are either supported on the boundary of the domain on which the dynamics evolves, or locally supported within said domain.

The objective of such controllability or reachability problems would be the steering of the controlled PDE solutions to a pre-assigned pro file or state.

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