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Feedback-invariant optimal control theory and differential geometry. I: Regular extremals. (English) Zbl 0952.49019
This paper is devoted to the unification of the theory of smooth optimal control problems and that part of differential geometry which deals with geodesics of different kinds.
Section 1 analyses the \({\mathcal L}\)-derivatives of smooth mappings. Section 2 realizes a connection between smooth control systems and basic structures of differential geometry. Section 3 gives the computation of \({\mathcal L}\)-derivative of the boundary-value mapping and studies the regular extremals (which are trajectories of a fixed Hamiltonian system). Section 4 introduces and investigates Jacobi curves as curves in a Lagrangian Grassmannian. Section 5 studies the canonical connections of Hamiltonian systems and of DEs of second-order. Section 6 finds explicit geometrical objects defined by two-dimensional control systems.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
53C22 Geodesics in global differential geometry
93B52 Feedback control
58E25 Applications of variational problems to control theory
53B05 Linear and affine connections
53B15 Other connections
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