DUALITY: DETECTABILITY VERSUS

STABILIZABILITY IN THE STOCHASTIC

CONTEXT^*

Vasile Drăgan

Abstract

The aim of this paper is to extend to the stochastic framework the well known duality relation between the detectability property and the stabilizability property of a finite dimensional, linear time invariant, deterministic system. We consider continuous time linear stochastic systems having the state space representation described by a system of Itˆo differential equations with periodic coefficients possibly subject to some stochastic perturbations modeled by a standard homogeneous Markov process with a finite number of states. It will be seen that in the case when the given system is affected by jump Markov perturbations a state space representation of the dual triple may be rigorously defined if and only if the transition probability matrix of the Markov process is double stochastic (in the sense that the sums of all elements from its each row and each column are equal 1).

Keywords: continuous time linear stochastic systems, stochastic detectability, stochastic stabilizability, duality, periodic coefficients.

MSC: 93E15, 93B05, 93B07, 93D15, 47B65.

DOI https://doi.org/10.56082/annalsarscimath.2024.2.287

Abstract Article Volume 16 no 2 / 2024