Le problème linéaire-quadratique pour les systèmes positifs

Abstract

The main goal of this work is the study of the infinite horizon positive linear quadratic problem. The objective, for a given positive system, is to find conditions such that the resulting LQ-optimal closed-loop system is positive. Firstly some concepts of the theory of nonnegative matrices are described. Then some properties of positive systems in continuous and discrete time are developed. Finally we present the positive LQ problem both in continuous and discrete time.\\ In order to obtain the existence of a solution in continuous time, we use a Newton-type iterative scheme converging to the unique stabilizing positive semi-definite solution for the algebraic Riccati equation for stabilisable and detectable systems. Then we study the positive LQ problem in discrete time and finite horizon, by using the minimum principle. We obtain necessary and sufficient conditions and also a positivity criterion which guarantee the existence of a solution. Lastly these results are extended to the infinite horizon problem as a limit of a sequence of finite horizon problems.

Date of Award	24 Aug 2006
Original language	French
Supervisor	Joseph Winkin (Supervisor)