# On the Model Motions in a Control Problem with Functional Constraints on Disturbances

D.A. SerkovA control problem for a system described by an ordinary differential equation is considered. It is suggested that the values of the control and of disturbance belong compact sets at every instant. It is also assumed that the disturbance meets some additional functional constraints showing the nature of the problem under consideration. The control quality is assessed by the functional continuous in the metrics of uniform convergence over the set of phase paths of the system. As it is previously stated, a strategy with full memory solves the control problem under compact constraints to the disturbance as well as under other functional constraints which are reduced to them. At the same time, the strategies constructed for the cases above are not universal, i.e. they depend on the starting position of the system motion. The question of possibility to solve the control problem with functional constraints in a narrower (classic) set of strategies (positional strategies) remains open. This paper gives the construction of the universal optimal strategy using a model of the control system in the feedback path. The examples that lead to the expansion of the class of solution strategies up to strategies with full memory are also given.Full text

- Keywords
- optimal guarantee, strategies with full memory, functional constraints.
- References
- 1. Krasovskii N.N., Subbotin A.I. Game-Theoretical Control Problems. N.Y., Springer-Verlag, 1988. 517 p.

2. Subbotin A.I., Chentsov A.G. Optimization of Guarantee in Control Problems [Optimizatsiya garantii v zadachah upravleniya]. Moscow, Nauka, 1981. 288 p.

3. Barabanova N.N., Subbotin A.I. On Continuous Evasion Strategies in Game Problems on the Encounter of Motions: PMM. Journal of Applied Mathematics and Mechanics, 1970, vol. 34, issue 5, pp. 796-803.

4. Barabanova N.N., Subbotin A.I. On Classes of Strategies in Differential Games of Evasion of Contact: PMM. Journal of Applied Mathematics and Mechanics, 1971, vol. 35, issue 3, pp. 385-395.

5. Kryazhimskii A.V. The Problem of Optimization of the Ensured Result: Unimprovability of Full-Memory Strategies. Constantin Caratheodory: An International Tribute, T.M. Rassias Ed., World Scientific. 1991.

6. Krasovskii N.N. Programm Absorption in Differential Games [Programmnoye pogloshcheniye v differentsialnykh igrakh]. Dokl. Acad. Nauk SSSR. [Reports to Academy of Science of USSR], 1971, vol. 201, no. 3, pp. 270-272.

7. Krasovskii N.N., Subbotin A.I. An Alternative for the Game Problem of Convergence: PMM. Journal of Applied Mathematics and Mechanics, 1970, vol. 34, issue 6, pp. 1005-1022.

8. Serkov D.А. Guaranteed Control under the Functional Restrictions on Disturbance [Garantirovannoye upravleniye pri funktsionalnykh ogranicheniyakh na pomekhu]. Matematicheskaya teoriya igr i yeye prilozheniya [Mathematical Game Theory and Its Applications], 2012, vol. 4, issue 2, pp. 71-95.

9. Warga J. Optimal Control of Differential and Functional Equations. N.Y., Academic Press, 1972. 544 p.

10. Kryazhimskii A.V., Osipov Yu.,S. On the Control Modeling in Dynamic System [O modelirovanii upravleniya v dinamicheskoy sisteme]. Izv. Acad. Nauk SSSR. Tehn. Kibernet. [Proceedings of the Academy of Sciences of the USSR. Tech. Cybernetics], 1983, vol. 2, pp. 51-60.

11. Osipov Yu.S., Kryazhimskii A.V. Inverse Problem of Ordinary Differential Equations: Dynamical Solutions. London, Gordon and Breach. 1995.

12. Subbotina N.N. Universal Optimal Strategies in Positional Differential Games [Universalnyye optimalnyye strategii v pozitsionnykh differentsialnykh igrakh]. Differentsial'nye uravneniya [Differential Equation], 1983, vol. 19, no. 11, pp. 1890-1896.

13. Chentsov A.G. Programming Constructs in Differential Games with Information Memory [Programmnyye konstruktsii v differentsialnykh igrakh s informatsionnoy pamyatyu]. Optimal Control of Systems with Uncertain Information [Optimalnoye upravleniye sistemami s neopredelennoy informatsiyey], Sverdlovsk, 1980, pp. 141-144.