IJPAM: Volume 37, No. 3 (2007)

THE FUNCTIONAL ANALYTIC SETTING OF HUM
PART I: GENERAL THEORY

Michael Pedersen
Department of Mathematics
Matematiktorvet - Building 303S
Technical University of Denmark
Kgs. Lyngby, DK 2800, DENMARK
e-mail: M.Pedersen@mat.dtu.dk


Abstract.The question of controllability for partial differential evolution equations of hyperbolic and parabolic nature has been studied intensively over the last decade, motivated and inspired by numerous applications in science and technology. The problems to be considered here are the boundary control problems that can be formulated as follows:

Let $u(x,t)$ denote the state of the ``system" as a function of space $x \in \Omega$, and of time $t$. We are allowed to act on the system by ``control variables''. These are functions $\kappa $ which are applied on $\Gamma $, the boundary of $\Omega$, or merely on parts of the boundary. The basic aim is to achieve exact controllability, i.e., given a time $T>0$ and initial data, find a control $\kappa $ that drives the system to a certain state at time $T$. In this paper we analyze the Hilbert Uniqueness Method - HUM - in terms of functional analysis and provide a unified setting in terms of reconstruction and controllability operators. In Part II of this paper we will then apply the functional analytic methods to the Mindlin-Timoshenko plate system.

Received: March 12, 2007

AMS Subject Classification: 35B37, 35L35

Key Words and Phrases: functional analysis, Hilbert uniqueness method, reconstruction and controllability operators, Mindlin-Timoshenko plate system

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 37
Issue: 3