The local solvability of a Hamilton-Jacobi-Bellman PDE around a nonhyperbolic critical point

We show the existence of a local solution to a Hamilton-Jacobi-Bellman (HJB ) PDE around a critical point where the corresponding Hamiltonian ODE is no t hyperbolic, i.e., it has eigenvalues on the imaginary axis. Such problems arise in nonlinear regulation, disturbance rejection, gain scheduling, and linear parameter varying control. The proof is based on an extension of th e center manifold theorem due to Aulbach, Flockerzi, and Knobloch. The meth od is easily extended to the Hamilton-Jacobi-Isaacs (HJI) PDE. Software is available on the web to compute local approximtate solutions of HJB and HJI PDEs.