A.A. Agrachev, R.V. Gamkrelidze, Symplectic geometry for optimal control

Journal title: Nonlinear controllability and optimal control
Year: 1990
Pages: 263–277
Authors: A.A. Agrachev, R.V. Gamkrelidze
Title: Symplectic geometry for optimal control

One of the main invariants of an extremal in a regular variational problem is its Morse index. If the extremal is optimal then its Morse index is zero. In the general case the Morse Index could be interpreted as the minimal number of (independent) additional relations that have to be satisfied by the admissible varioations of the given trajectory in order to make it optimal.
It turns out that there an analogue of this index in optimal control. If the set of control parameters is open then the index is easy to define, though much harder to compute. For strongly nondegenerate cases the analogue of the Morse formula was obtained in [8] and [10]. The systematic use of symplectic geometry offers a different way of computing the index, stable under practically any perturbation, thus closing the problem for singular extremals, cf. [4], [5].
In this paper we employ the latter method and compute the index for the problem with constraints on the control parameters. We consider in some detail bang-bang controls and then briefly present a universal formula valid for bang-bang as well as singular parts of the optimal trajectory.
We apply our results to the study of optimal control problems for smooth systems. Necessary conditions for optimality are formulated for controls (including bang-bang controls) that satisfy Pontryagin's Maximum Principle. As an example, we consider the case of a rigid body which is controlled by roating it with a given velocity around two fixed axes.

Year: 1990
Number of pages: 263-277
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