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The Liouville theorem in optimal control problems. (English. Russian original) Zbl 0861.70017

Differ. Equations 30, No. 11, 1808-1814 (1994); translation from Differ. Uravn. 30, No. 11, 1958-1965 (1994).
The extremals of a controlled completely integrable dynamic system are interpreted as integral trajectories of a piecewise smooth Hamiltonian system on a symplectic manifold. The author studies the problem of sewing such Hamiltonian systems from the set of the smooth ones on manifolds with edges. The passing of the extremal through the synthesis hypersurface corresponds to transition of this extremal from one edge to another in the sewing procedure. The main theorem establishes the equivalence of completely integrable piecewise smooth Hamiltonian systems to the smooth Liouville integrable Hamiltonian systems. An illustrative example is given for piecewise smooth Hamiltonian system in the one-dimensional case \(\dot x=u\).

MSC:

70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics
70H05 Hamilton’s equations
93C15 Control/observation systems governed by ordinary differential equations
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