×

Symplectic geometry for optimal control. (English) Zbl 0719.49023

Nonlinear controllability and optimal control, Lect. Workshop, New Brunswick/NJ (USA) 1987, Pure Appl. Math., Marcel Dekker 133, 263-277 (1990).
[For the entire collection see Zbl 0699.00040.]
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 variations of the given trajectory in order to make it optimal.
It turns out that there is 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 by M. R. Hestenes [Calc. Var. Control Theory, Proc. Symp. Math. Res. Cent., Madison 1975, 289-304 (1976; Zbl 0344.49003)] and A. V. Sarychev [Math. USSR, Sb. 41, 383-401 (1982); translation from Mat. Sb., Nov. Ser. 113(155), 464- 486 (1980; Zbl 0484.49012)]. 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. the authors, Sov. Math., Dokl. 33, 392-395 (1986); translation from Dokl. Akad. Nauk SSSR 287, 521-524 (1986; Zbl 0614.58016); and the first author, “Quadratic mappings in geometric control theory” (Russian), Itogi Nauki Tekh., Ser. Probl. Geom. 20, 111-205 (1988); English translation to appear in J. Sov. Math.
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 rotating it with a given velocity around two fixed axes.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)