Pukhlikov, A. V. 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 \(\dot x=u\). Reviewer: V.Chernyatin (Szczecin) Cited in 2 Documents 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 Keywords:sewing problem; piecewise smoothness; controlled completely integrable dynamic system; integral trajectories; symplectic manifold; manifolds with edges; extremal PDFBibTeX XMLCite \textit{A. V. Pukhlikov}, Differ. Equations 30, No. 11, 1808--1814 (1994; Zbl 0861.70017); translation from Differ. Uravn. 30, No. 11, 1958--1965 (1994)