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Manifold-following approximate solution of completely hypersensitive optimal control problems. (English) Zbl 1346.49040
Summary: The solution to a completely hypersensitive optimal control problem may shadow trajectories on the stable and unstable manifolds of an equilibrium point in the state-costate phase space. If such shadowing occurs, the solution of the Hamiltonian boundary value problem, that constitutes the first-order necessary conditions for optimality, can be approximated as a composite of an initial segment on the stable manifold leading to the equilibrium point and a final segment on the unstable manifold departing from the equilibrium point. Using a dichotomic basis, the Hamiltonian vector field can be decomposed into stable and unstable components, and the unspecified boundary conditions for the initial and terminal segments can be determined such that the initial and final conditions are, respectively, on the stable and unstable manifolds of the equilibrium point. In this paper, we propose and justify the use of finite-time Lyapunov vectors to construct an approximate dichotomic basis and develop a corresponding manifold-following solution approximation method. The method is illustrated on two examples and shown to be more accurate than a similar method that uses eigenvectors of the frozen-time linearized dynamics.

MSC:
49M05 Numerical methods based on necessary conditions
49K15 Optimality conditions for problems involving ordinary differential equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
34N05 Dynamic equations on time scales or measure chains
37D05 Dynamical systems with hyperbolic orbits and sets
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
Software:
BNDSCO; GPOPS
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References:
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