Systems & Control Letters 58 (2009) 254–258 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Necessary and sufficient conditions for a solution to the risk-sensitive Poisson equation on a finite state space ✩ Rolando Cavazos-Cadena a , Daniel Hernández-Hernández b,∗ a Departamento de Estadística y Cálculo, Universidad Autónoma Agraria Antonio Narro, Buenavista, Saltillo, COAH 25315, Mexico b Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato, GTO 36000, Mexico article info Article history: Received 1 February 2008 Received in revised form 21 October 2008 Accepted 2 November 2008 Available online 6 December 2008 Dedicated to Professor O. Hernández-Lerma, on the occasion of his sixtieth birthday Keywords: Unique recurrent class Hitting time Constant average cost Unichain property Multiplicative poisson equation abstract A Markov chain with finite state space endowed with a cost function is considered. The transition mechanism is stationary, the observer has a constant risk-sensitivity, and the overall performance of the chain is measured by the risk-sensitive long-run average cost. In this context, the existence of solutions of the corresponding Poisson equation for arbitrary cost function is characterized in terms of the communication properties of the transition matrix. The result in this direction establishes that the Poisson equation has a solution for each cost function if, and only if, the transition matrix has a unique recurrent class and a strong form of the Doeblin condition holds. © 2008 Elsevier B.V. All rights reserved. 1. Introduction This work concerns the risk-sensitive average cost function associated with a Markov chain on a finite state space endowed with a cost structure, and the main objective is to determine necessary and sufficient conditions on the transition matrix so that, for an arbitrary cost function, the corresponding risk-sensitive Poisson equation has a solution. To describe this problem in a more precise manner, let {X n } be a Markov chain evolving on a finite space S , and let [p x,y ] x,y∈S be the (time-invariant) transition matrix. It is assumed that the observer has a constant and non-null risk sensitivity λ, and that the overall performance of the chain starting at each state x ∈ S is measured by the corresponding long-run average cost J C (x): ✩ This work was supported by the PSF Organization under Grant No. 07-01, and in part by CONACYT under Grants 25357 and 61423. ∗ Corresponding address: Centro de Investigacion en Matematicas, Callejon Jalisco s/nApartado Postal 402, 36000 Guanajuato, GTO, Mexico Tel.: +52 473 7327155; fax: +52 473 7325749. E-mail addresses: rcavazos@narro.uaaan.mx (R. Cavazos-Cadena), dher@cimat.mx (D. Hernández-Hernández). J C (x): = lim sup n→∞ 1 nλ log   E   e λ n−1 ∑ k=0 C (X k )       X 0 = x     . (1.1) The characterization of J C (·) is usually based on the following Poisson equation, where g is real number and h(·) is a real function on S : e λg +h(x) = e λC (x)  y∈S p xy e h(y) , x ∈ S . (1.2) When this equality holds it follows that J C (·) ≡ g [1,2]. In the analysis of controlled Markov chains, an optimality equation similar to (1.2) arises as a vehicle to determine the optimal risk- sensitive average cost, as well as the decision rule allowing to achieve the minimum cost [3–7]. On the other hand, the existence of a pair (g , h(·)) solving (1.2) is guaranteed if the transition matrix [p x,y ] x,y∈S is aperiodic and the state space is a communicating class (see [8]); indeed, in this case it follows from the Perron–Frobenious theorem [9] that (1.2) holds if e λg is the largest eigenvalue of [e λC (x) p xy ] xy∈S with e h(·) as a positive eigenvector; an alternative approach to this result, based on the risk-sensitive total cost index and avoiding the aperiodicity assumption, was given in [5]; see also [10,11]. However, if the class of transient states is nonempty, J C (·) 0167-6911/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2008.11.001