Exact slow-fast decomposition of the Hamilton-Jacobi equation of singularly perturbed systems

We study a Hamilton-Jacobi partial differential equation, arising in an optimal control problem for an affine nonlinear singularly perturbed system. This equation is solvable if there exists a special invariant manifold of the corresponding Hamiltonian system. We obtain exact slow-fast decomposition of the Hamiltonian system and of the special invariant manifold into the slow and the fast ones. We get sufficient conditions for the solvability of the Hamilton-Jacobi equation in terms of the reduced-order slow submanifold, or in the hyperbolic case, in terms of a reduced-order slow Riccati equation. On the basis of this decomposition we construct asymptotic expansions of the optimal state-feedback, optimal trajectory and optimal open-loop control in the powers of a small parameter