Approximation of Infinite-Dimensional Systems – AIDS

Development of tools and methods for approximation of infinite dimensional systems.

The results we obtained are concerned with
1. Synthesis of symplectic scheme for discretization of port hamiltonian systems.
2. Approximation of distributed delays.
3. Analysis of systems governed by difference equations.
4. Control of distributed parameter systems by pole placement.
5. Approximation of distributed systems in the graph topology.

The prospectives of this work are
1. Generalization of symplectic schemes for distributed parameter systems in hamiltonian framework.
2. Applicability of systems governed by difference equations for the synthesis of numerical schemes of PDE.
3. Development of efficient numerical methods for approximation in graph topology.
4. Link with discretization methods in an input-output framework..

1 publication submitted in journal
2 publications submitted in international conference
5 publications under construction for international conferences

In this project, we address the approximation problem of distributed parameter systems, for analysis and control purposes.

We are interested in the class of distributed parameter systems governed by linear or quasi-linear partial differential equations, with appropriate boundary and initial conditions. From an input-output approach, these dynamical systems are operators with input signals and output signals, related respectively to control inputs, disturbances, and measurements or controlled variables. These input and output signals are time and space functions, and can be either localized or distributed in time and/or space domains.

To exploit a dynamical model for analysis, simulation or control purposes, the analytical representation of the distributed parameter system has to be simpli?ed by an appropriate approximation method. Our work is based on the operator approximation, and is decomposed in two parts. The ?rst part concerns the realization of an approximation scheme by spatial discretization using Hamiltonian formulation. The second part develops an approximation method by a class of linear time-delay operators.

We highlight additional requirements on these two approximation methods to conserve structure properties, like energy balance for the Hamiltonian approach, or weak-strong controllability for the time-delay approach. These two approaches will make use of common algebraic tools.
From these approximations, some control issues for distributed parameter systems are discussed.

With this basic research, we aim at working out new andanalysis and the control of distributed parameter systems.