Abstract
Deterministic optimal impulse control problem with terminal state constraint is considered. Due to the appearance of the terminal state constraint, the value function might be discontinuous in general. The main contribution of this paper is the introduction of an intrinsic condition under which the value function is proved to be continuous. Then by a Bellman dynamic programming principle, the corresponding Hamilton-Jacobi-Bellman type quasi-variational inequality (QVI, for short) is derived. The value function is proved to be a viscosity solution to such a QVI. The issue of whether the value function is characterized as the unique viscosity solution to this QVI is carefully addressed and the answer is left open challengingly.
Highlights
It is well-known that in general classical continuous-time optimal control theory, there are two major approaches: variational method leading to Pontryagin’s maximum principle (MP, for short), and dynamic programming method leading to Hamilton-Jacobi-Bellman (HJB, for short) equation
The latter usually works for the problems without terminal state constraint and it leads to a characterization of the value function as the unique viscosity solution to the HJB equation, formally, optimal control of state feedback form can be obtained [1, 5, 17, 23, 39]
We have introduced an intrinsic condition under which, together with other routine conditions, the value function of the optimal impulse control with terminal state constraint is continuous
Summary
It is well-known that in general classical continuous-time optimal control theory, there are two major approaches: variational method leading to Pontryagin’s maximum principle (MP, for short), and dynamic programming method leading to Hamilton-Jacobi-Bellman (HJB, for short) equation The former could work for the problems with possible terminal state constraint and it gives necessary conditions for (possibly existed) open-loop optimal controls [27, 39]. Due to the presence of the terminal state constraint, the value function of the optimal impulse control problem is proved to be locally Holder (or Lipschitz) continuous only, even under our discovered condition, and it could grow at least linearly (no slower than the growth of the impulse cost) These properties essentially prevent us from directly using the current available techniques to prove the value function being the uniqueness of viscosity solution to the corresponding HJB QVI.
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