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 not 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 the center manifold theorem due to Aulbach, Flockerzi, and Knobloch. The method 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.