arXiv subjects
Publications and source records attributed to Andrei Dmitruk.
We consider the time-optimal problem for a classical system of "double integrator" under the presence of a linear state constraint. By using the Maximum Principle of Dubovitskii and Milyutin, we determine a complete synthesis of optimal trajectories and qualitatively analyze the corresponding Lagrange multipliers.
We consider some critical claims concerning our above paper, and reply to these claims.
We consider a class of optimal control problems with a state constraint and investigate a trajectory with a single boundary interval (subarc). Following R.V. Gamkrelidze, we differentiate the state constraint along the boundary subarc, thus reducing the original problem to a problem with mixed control-state constraints, and show that this way allows one to obtain the full system of stationarity conditions in the form of A.Ya. Dubovitskii and A.A. Milyutin, including the sign definiteness of the measure (state constraint multiplier), i.e., the nonnegativity of its density and atoms at junction points. The stationarity conditions are obtained by a two-stage variation approach, proposed in this paper. At the first stage, we consider only those variations, which do not affect the boundary interval, and obtain optimality conditions in the form of Gamkrelidze. At the second stage, the variations are concentrated on the boundary interval, thus making possible to specify the stationarity conditions and obtain the sign of density and atoms of the measure.
We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packages of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is "compressed" in one universal optimality condition by using the concept of centered family of compacta.
We consider a problem on maximizing the height of vertical flight of a material point ("meteorological rocket") in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. The original Goddard problem is simplified by removing the dependence on the rocket mass from the equations of motion. Using the maximum principle we find all possible types of Pontryagin extremals and classify them w.r.t. problem parameters. Since the velocity of the point can be negative, we obtain some new types of extremals with two or three switching points, which optimality should be further investigated.