TY - JOUR
T1 - Exact slow-fast decomposition of a class of non-linear singularly perturbed optimal control problems via invariant manifolds
AU - Fridman, F.
PY - 1999
Y1 - 1999
N2 - We study a Hamilton-Jacobi partial differential equation, arising in an optimal control problem for an affine non-linear singularly perturbed system. This equation is solvable iff 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 sufficient 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.
AB - We study a Hamilton-Jacobi partial differential equation, arising in an optimal control problem for an affine non-linear singularly perturbed system. This equation is solvable iff 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 sufficient 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.
UR - http://www.scopus.com/inward/record.url?scp=0033354175&partnerID=8YFLogxK
U2 - 10.1080/002071799220074
DO - 10.1080/002071799220074
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AN - SCOPUS:0033354175
SN - 0020-7179
VL - 72
SP - 1609
EP - 1618
JO - International Journal of Control
JF - International Journal of Control
IS - 17
ER -