Quadratic costs and second moments of jump linear systems with general Markov chain

This paper presents an analytic, systematic approach to handle quadratic functionals associated with Markov jump linear systems with general jumping state. The Markov chain is finite state, but otherwise general, possibly reducible and periodic. We study how the second moment dynamics are affected by the additive noise and the asymptotic behaviour, either oscillatory or invariant, of the Markov chain. The paper comprises a series of evaluations that lead to a tight two-sided bound for quadratic cost functionals. A tight two-sided bound for the norm of the second moment of the system is also obtained. These bounds allow us to show that the long-run average cost is well defined for system that are stable in the mean square sense, in spite of the periodic behaviour of the chain and taking into consideration that it may not be unique, as it may depend on the initial distribution. We also address the important question of approximation of the long-run average cost via adherence of finite horizon costs. Research supported by FAPESP Grants 03/06736-7 and 04/06947-0 and by the CNPq Grants 304429/2007-4, 304856/2007-0 and 306466/2010-4. E. F. Costa (B) Universidade de São Paulo (USP), Depto. de Matemática Aplicada e Estatística, C.P. 668, São Carlos, SP 13560-970, Brazil e-mail: [email protected] A. N. Vargas · J. B. R. do Val Universidade Estadual de Campinas (UNICAMP), Fac. de Eng. Elétrica e de Computação, Depto. de Telemática, C.P. 6101, Campinas, SP 13081-970, Brazil e-mail: [email protected] J. B. R. do Val e-mail: [email protected]