Exact slow± fast decomposition of a class of non-linear singularly perturbed optimal control problems via invariant manifolds

We study a Hamilton± Jacobi partial di€ erential equation, arising in an optimal control problem for an a ne non-linear singularly perturbed system. This equation is solvable i€ 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 slow and fast components. We get su cient conditions for the solvability of the Hamiltonian± 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 powers of a small parameter.