ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

17. Control Theory and Optimization

1991 MS Classification: 49 Borisov, Vladimir F., Moscow State (Lomonosov) University, Moscow, Russia Bundles with chattering extremals Pontryagin’s maximum principle for affine in control problem leads to the following discontinuous Hamiltonian (2n)–system

ψ̇ = −∂H/∂x, ẋ = ∂H/∂ψ,

where H = H_{0}(ψ, x) + uH_{1}(ψ, x), u = sgn H_{1}(x). For generic systems possessing a singular extremal of order q ≥ 3 we prove that there exist [q/2] (2n − 2q + 2)–dimensional bundles with the common (2n − 2q)–dimensional base and two–dimensional piece–wise smooth fibres such that each fibre consists of chattering extremals, i. e. trajectories having a countable number of switches at finite time. The result is a generalisation of a theorem on bundles proved in [1] for systems possessing singular extremals of order q = 2. [1]. Zelikin, M. I., Borisov, V. F. Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering. Birkhäuser, Boston, 1994.