On the Bellman equation for infinite horizon problems with unbounded cost functional (Q1964693)

The author investigates a class of nonlinear infinite horizon optimal control problems minimizing a cost functional of the type \[ J(x,\alpha)=\int_0^\infty l(y_x(t),\alpha(t))e^{-\lambda t}\, dt. \] Since the discount rate \(\lambda\) is only required to be nonnegative and the set of control values \(A\) is allowed to be unbounded this class in particular includes the linear quadratic (LQ) control problem. Under standard regularity assumptions on the problem data it is shown that the optimal value function \(V\) of this problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation \[ \lambda V(x) + \sup_{a\in A}\{-f(x,a)DV(x)-l(x,a) \}=0. \] In addition, the regularity of \(V\), the dependence of \(V\) on the problem data (in terms of stability of the solution to the HJB equation) and the behaviour of optimal controls for the LQ problem under perturbations are addressed.