Optimal stochastic-control of linear stochastic-systems with Poisson-process disturbances

We obtain results similar to those for linear–quadratic gain problems on the control system structure for optimal linear–quadratic regulator problems with Poisson noise disturbance. If the coefficient matrices of the system dynamics and the performance index are time-invariant, the optimal state and optimal control of the finite time problem, respectively, converge to those of the infinite time problem quasi-uniformly, almost surely. Both the long term average cost criterion and the discounted cost criterion are investigated by use of the stochastic version of Gronwall inequality for infinite time problems.