Remarks on Viscosity Super-Solutions of Quasi-Variational Inequalities

2020/10/24 by Zhou, Yue, Feng, Xinwei, Yong, Jiongmin
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2010.12738

Abstract

For Hamilton-Jacobi-Bellman (HJB) equations, with the standard definitions of viscosity super-solution and sub-solution, it is known that there is a comparison between any (viscosity) super-solutions and sub-solutions. This should be the same for HJB type quasi-variational inequalities (QVIs) arising from optimal impulse control problems. However, according to a natural adoption of the definition found in Barles 1985, Barles 1985b, the uniqueness of the viscosity solution could be guaranteed, but the comparison between viscosity super- and sub-solutions could not be guaranteed. This paper introduces a modification of the definition for the viscosity super-solution of HJB type QVIs so that the desired comparison theorem will hold.

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