Kim Ung, a section head at the Faculty of Applied Mathematics, has studied a deterministic infinite horizon, a mixed continuous and impulse control problem with unbounded cost functional.
Major difficulties arise from the unboundedness of value functions.
He proved the comparison result of viscosity solution by converting unbounded value functions into bounded ones by suitable transformation and proved that the value function is the unique viscosity solution of the related first order Hamilton-Jacobi quasi-variational inequality (QVI). He constructed a time-discretization scheme for the QVI and proved that an approximate value function exists and it is the unique solution of the approximate QVI.
He also proved that the solution of the time discrete scheme converges to the viscosity solution of the QVI, when the discretization step goes to zero, and that 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.
You can find more information in his paper “Approximation Schemes for Impulse Control Systems with Unbounded Cost Functional” in “Proceedings of KUTIC-2025”.