A power penalty method for discrete HJB equations

Abstract

We develop a power penalty approach to the discrete Hamilton–Jacobi–Bellman (HJB) equation in $$ \mathbb {R}^N $$ in which the HJB equation is approximated by a nonlinear equation containing a power penalty term. We prove that the solution to this penalized equation converges to that of the HJB equation at an exponential rate with respect to the penalty parameter when the control set is finite and the coefficient matrices are M-matrices. Examples are presented to confirm the theoretical findings and to show the efficiency of the new method.