Reeb foliation in the optimal control problem with phase constraints. (English. Russian original) Zbl 1290.49039
J. Math. Sci., New York 177, No. 2, 229-243 (2011); translation from Sovrem. Mat. Prilozh. 69 (2011).
Summary: An optimal control problem with two phase constraints and a two-dimensional control is considered. We prove that its optimal trajectories have a countable number of points of contact with the phase constraint bound on a finite time interval. The optimal synthesis of the problem after applying the quotient mapping by the action of the scale group has a complicated topological structure and, in some neighborhood of singular extremals, it is the Reeb foliation.
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
References:
| [1] | V. V. Dikusar and A. A. Milyutin, Qualitative and Numerical Methods in Maximum Principle [in Russian], Nauka, Moscow (1989). · Zbl 0704.65048 |
| [2] | M. I. Zelikin, ”Synthesis of optimal trajectories that define the Reeb foliation,” Proc. Steklov Inst. Math., 233, 81–86 (2001). · Zbl 1014.49029 |
| [3] | M. I. Zelikin, V. V. Gael, ”Accumulation of tangent points with the bound and Lagrangian manifolds in problems with phase constraints,” J. Math. Sci., 177, No. 2, 299–328 (2011). · Zbl 1290.49071 · doi:10.1007/s10958-011-0458-8 |
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