The paper considers some control problems for systems described on infinite-dimensional spaces. A mathematical framework in terms of semigroups is developed which enables the generalisation of the finite-dimensional results to infinite dimensions, and which includes partial differential equations and delay equations as special cases. After first surveying some finite-dimensional results, abstract dynamical systems are introduced and the systems theory concepts of stability, controllability, observabiity and the linear quadratic problem are analysed. Throughout the paper the abstract theory is applied to a number of examples to obtain specific results.

Infinite-Dimensional Linear Systems Theory

The controllability and linear quadratique optimal control problems of time-varying discrete distributed systems are analysed. We show that, under certain hypothesis, the Hilbert Uniqueness Method (HUM) can be used to solve these problems. Examples and numerical simulations are given.

Controllability and optimal control problem for linear time-varying discrete distributed systems

A linear quadratic optimal control problem for a class of discrete distributed systems is analyzed. To solve this problem, we introduce an adequate topology and establish that optimal control can be determined though an inversion of the appropriate isomorphism. An example and a numerical approach are given.

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A quadratic optimal control problem for a class of linear discrete distributed systems

Linear Quadratic Control Problem With Fixed Final State for Discrete-Time Distributed Systems

We consider a class of linear discrete-time systems controlled by a continuous time input. Given a desired final state xd, we investigate the optimal control which steers the system, with a minimal cost, from an initial state x0 to xd. We consider both discrete distributed systems and finite dimensional ones. We use a method similar to the Hilbert Uniqueness Method (HUM) to determine the control and the Galerkin method to approximate it, we also give an example to illustrate our approach.

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On the Controllability of a Class of Discrete Distributed Systems

This paper deals with the exact controllability and the quadratic cost control problems for distributed systems with a discrete delayed control. The Hilbert uniqueness method is developed to express the solution of both problems. To illustrate the theory an example of diffusion system is treated.

Exact controllability and optimal control for distributed systems with a discrete delayed control

The present work deals with the linear quadratic control problem for a discrete distributed system with terminal convex constraint. Using techniques of perturbation by feedback, it is shown that the resolution of the considered problem is equivalent to that of a controllability, one so-called Extended Exact Controllability with time-varying operators. The Hilbert uniqueness method approach is then extended to this case to provide an explicit form for the optimal control. In the same framework, the inequality constraint case is examined for which a practical numerical resolution is given. Finally, the results obtained are used to treat a minimum-time reachability problem.

Abdelatif Ouansafi

Papers

Exact controllability and optimal control for distributed systems with a discrete delayed control

Linear quadratic control problem with a terminal convex constraint for discrete-time distributed systems