On the Bellman equation for some unbounded control problems

We consider a class of finite horizon optimal control problems with unbounded data for nonlinear systems, which includes the Linear-Quadratic (LQ) problem. We give comparison results between the value function and viscosity sub- and supersolutions of the Bellman equation, and prove uniqueness for this equation among locally Lipschitz functions bounded below. As an application we show that an optimal control for the LQ problem is nearly optimal for a large class of small unbounded nonlinear and non-quadratic perturbations of the same problem.