Ivan Samylovskiy

On two approaches to necessary conditions for an extended weak minimum in optimal control problems with state constraints

We consider a class of optimal control problems with a scalar state constraint. For a trajectory with a single boundary subarc, we first obtain, using a special technique (two-stage var iation approach), optimality conditions in the form of Gamkrelidze, and then obtain the full set of optimality conditions in the Dubovitskii-Milyutin form, including the nonnegativity of the measure density and its atoms at the junction points.

Optimal trajectories in a maximal height problem for a simplified version of the Goddard model in case of generalized media resistance function

We consider a family of the problems on maximization of the height of the vertical flight of a material point in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. Using the maximum principle we obtain classification of trajectories (w.r.t. problem parameters) which are suspected to be optimal and check that for some classical rocket systems optimal control in our model is the classical bang-bang or bang-singular-bang one. We obtain some new types of “potentially-optimal” trajectories which should be investigated.

On a trolley-like problem in the presence of a nonlinear friction and a bounded fuel expenditure

We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.

Optimal Synthesis in the Reeds and Shepp Problem with Onesided Variation of Velocity

We consider a time-optimal problem for the Reeds and Shepp model describing a moving point on a plane, with a onesided variation of the speed and a free final direction of the velocity. Using the Pontryagin Maximum Principle, we obtain all possible types of extremal and, analyzing them and discarding nonoptimal ones, construct the optimal synthesis.