Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8612Research ArticleOn Some Classes of Optimal Control Problem with State ConstraintsGorbachevaA VDepartment of nonlinear analysis and optimization; Department of applied mathematics Russian state social university 4, Wilhelm Pieck str., Moscow, Russia, 129226avgorbacheva@inbox.ruKaramzinD YuDmitry_karamzin@mail.ruPeoples’ Friendship University of RussiaDorodnicyn Computing Centre of the Russian Academy of Science150120161111808092016Copyright © 2016,2016A Borel measure Lagrange multiplier appears in the maximum principle for state constrained problems. The question of continuity or absolute continuity of the measure-multiplier is highly relevant for various applications in particular for some problems of kinematic control. The velocity in such problems is considered as a state variable. As soon as the magnitude of the velocity is bounded, for instance above, (which is quite natural in problems of kinematic control), this leads to the state constraints and to a measure Lagrange multiplier in the necessary optimality conditions. In Control Theory, the methods that are use to solve these conditions often require the continuity of the measure. In this paper, we consider some examples of optimal control problems with state constraints for which one can ensure that this measure is continuous, without a calculation of extremal process.optimal controlmaximum principlestate constraintsBorel measureHölder conditionоптимальное управлениепринцип максимумафазовые ограниченияборелевская мераусловие Гельдера[Arutyunov A.V., Karamzin D.Y. On Some Continuity Properties of the Measure Lagrange Multiplier from the Maximum Principle for State Constrained Problems // SIAM Journal on Control and Optimization. - 2015. - Vol. 53, No 4. - Pp. 2514-2540.][Arutyunov A.V. Optimality Conditions: Abnormal and Degenerate Problems. - Dordrecht/Boston/London: Kluwer Academic Publisher, 2000.][Arutyunov A.V., Karamzin D.Y., Pereira F.L. The Maximum Principle for Optimal Control Problems with State Constraints by R.V. Gamkrelidze: Revisited // J. Optim. Theory Appl. - 2011. - Vol. 149. - Pp. 474-493.][Zakharov E.V., Karamzin D.Y. On the Study of Conditions for the Continuity of the Lagrange Multiplier Measure in Problems with State Constraints // Differential Equations. - 2015. - Vol. 51, No 3. - Pp. 399-405.]