Volume 27, 2021
|Number of page(s)||36|
|Published online||08 December 2021|
Continuity of the value function for deterministic optimal impulse control with terminal state constraint
School of Mathematics and Statistics, Central South University,
2 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, China.
3 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.
* Corresponding author: email@example.com
** This author is supported by National Natural Science Foundation of China (No. 12001317), Shandong Provincial Natural Science Foundation (No. ZR2020QA019) and QILU Young Scholars Program of Shandong University.
*** This author is supported in part by NSF Grant DMS-1812921.
Accepted: 18 November 2021
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.
Mathematics Subject Classification: 49N25 / 49L20 / 49L25
Key words: Optimal impulse control / terminal state constraint / continuity of value function / Hamilton-Jacobi-Bellman quasi-variational inequality / viscosity solution
© The authors. Published by EDP Sciences, SMAI 2021
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