International Journal Of Chemistry, Mathematics And Physics(IJCMP)

The impulse control problem with unbounded cost functional is different from the bounded case. We prove comparison result of viscosity solution by converting the unbounded value function into bounded one by suitable transformation and prove that the value function is the unique viscosity solution of the related first order Hamilton–Jacobi quasi-variational inequality (QVI). We construct a time discretization scheme for the QVI and prove that the approximate value function exists, that it is the unique solution of the approximate QVI. We also prove that the solution of the time discretization scheme converges to the viscosity solution of the QVI, when the discretization step goes to zero. The optimal control of the time discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the impulse control problem.