an:01431841
Zbl 0952.49019
Agrachev, A. A.; Gamkrelidze, R. V.
Feedback-invariant optimal control theory and differential geometry. I: Regular extremals
EN
J. Dyn. Control Syst. 3, No. 3, 343-389 (1997).
00065025
1997
j
49K15 53C22 93B52 58E25 53B05 53B15
smooth optimal control; geodesics; regular extremals; canonical connections; Hamiltonian systems
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.
Constantin Udri??te (Bucure??ti)