Hamilton–Jacobi–Bellman Quasi-Variational Inequality arising in an environmental problem and its numerical discretization

A Hamilton–Jacobi–Bellman Quasi-Variational Inequality (HJBQVI) for a river environmental restoration problem with wise-use of sediment is formulated and its mathematical properties are analyzed. A finite difference scheme with a penalization technique is then established for solving the HJBQVI. The scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions. A demonstrative application example of the HJBQVI is finally presented.