Appl Math Optim 45:45–61 (2002)
2002 Springer-Verlag New York Inc.
On Risk-Sensitive Ergodic Impulsive Control of Markov Processes
Institute of Mathematics, Polish Academy of Sciences,
Sniadeckich 8, 00-950 Warsaw, Poland
Warsaw School of Management and Marketing,
Al. Jerozolimskie 202, 02-486 Warsaw, Poland
Abstract. Impulsive control of continuous-time Markov processes with risk-
sensitive long-run average cost is considered. The most general impulsive control
problem is studied under the restriction that impulses are in dyadic moments only.
In a particular case of additive cost for impulses, the impulsive control problem is
solved without restrictions on the moments of impulses.
Key Words. Impulsive control, Risk-sensitive long-run average cost, Controlled
Markov processes, Bellman equation.
AMS Classiﬁcation. Primary 93E20, Secondary 93C40, 60J25.
Assume that on a probability space (, F, F
, P) we are given a time homogeneous
c`adl`ag Markov process X = (x
) taking values on a complete separable metric space
E, endowed with the Borel σ -algebra E. Moreover, assume that (x
) has a transition
, ·) at generic time t.
Consider the following impulsive control problem. At a Markov time τ the process
) can be shifted to an F
-measurable, U (x
)-valued random variable ξ, where U is
a compact-valued lower semicontinuous multifunction. An impulsive control strategy
V = (τ
) consists of sequences of strictly increasing Markov times τ
, i = 1, 2 ...,adapted to observation up to time τ
, such that ξ
∈ U (x
where by x
we denote the state of the controlled process just before the shift at time
. To describe an evolution of the controlled process (x
) under the strategy V we have
The work by the second author was supported by KBN Grant No. 2 P03A 01515.