Journal of Applied Mathematics and Stochastic Analysis
Volume 15 (2002), Issue 1, Pages 1-21

A weak convergence approach to hybrid LQG problems with infinite control weights

G. George Yin1 and Jiongmin Yong2

1Wayne State University, Department of Mathematics, Detroit 48202, MI, USA
2Fudan University, Laboratory of Mathematics for Nonlinear Sciences, Department of Mathematics, Shanghai 200433, China

Received 1 June 2001; Revised 1 December 2001

Copyright © 2002 G. George Yin and Jiongmin Yong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This work is concerned with a class of hybrid LQG (linear quadratic Gaussian) regulator problems modulated by continuous-time Markov chains. In contrast to the traditional LQG models, the systems have both continuous dynamics and discrete events. In lieu of a model with constant coefficients, these coefficients vary with time and exhibit piecewise constant behavior. At any time t, the system follows a stochastic differential equation in which the coefficients take one of the m possible configurations where m is usually large. The system may jump to any of the possible configurations at random times. Further, the control weight in the cost functional is allowed to be indefinite. To reduce the complexity, the Markov chain is formulated as singularly perturbed with a small parameter. Our effort is devoted to solving the limit problem when the small parameter tends to zero via the framework of weak convergence. Although the limit system is still modulated by a Markov chain, it has a much smaller state space and thus, much reduced complexity.