Convergence of implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities
For researchers in numerical analysis and optimal control, this work provides theoretical justification for existing computational methods, though it is an incremental theoretical contribution.
This paper provides rigorous convergence proofs for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities, closing a gap left by prior work. The proofs rely on a new notion of nonlocal consistency and a comparison principle.
In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 1341-1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles-Souganidis type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.