On the smoothness of solutions of linear-quadratic regulator for degenerate diffusions

The paper studies the smoothness of solutions of the degenerate Hamilton–Jacobi–Bellman (HJB) equation associated with a linear-quadratic regulator control problem. We establish the existence of a classical solution of the degenerate HJB equation associated with this problem by the technique of viscosity solutions, and hence derive an optimal control from the optimality conditions in the HJB equation.