Let denote the risk-sensitive discounted cost function defined by
The problem is to find an admissible control that minimizes .
We now specify the production constraints. For each , let
With this definition, the production constraint at time t is
We assume the demand rate z(t) to be a bounded process independent of .
We say that a control is admissible if u(t) is a -adapted measurable process and for all . Then our control problem can be written as follows:
Let . We consider the following control space:
and two control problems and defined as follows:
and
It can be seen below that when is small, can be approximated by and can be approximated further by. Therefore, can be approximated by . Then, a near optimal control for will be used to construct controls for that are nearly optimal.
Theorem 6.1 There exist constants and C such that, for ,
We show that can be approximated by and the value function of is a viscosity solution to the Isaacs equation of a zero-sum, two-player differential game. To simplify the notation, we take and consider the following control problem .
Theorem 6.2is a monotone increasing function of and
For each .
We write v0,0(x) as the value function of with the initial value x0=x. Note that for any random variable . Let such that and let denote a compact subset of Rn. We consider functions () that are right-continuous and have left-hand limits. Let denote the metric space of such functions that is equipped with the Skorohod topology .
We assume a.s. and for each and any , .
Theorem 6.3 v0,0(x) is the only viscosity solution to the following Isaacs equation:
Theorem 6.4 The following
assertions hold:
(i)
(ii) Letdenote a stochastic open loop -optimal control for , i.e.,
Let , where 1A denotes the indicator of a set A. Then, and
(iii) Letdenote a feedback -optimal control for , i.e.,Let
Assume that U(z,x) is locally Lipschitz in x, i.e., for some k>0,
Then, and