An application of Pontryagin's maximum principle in a linear quadratic differential game.

This paper deals with a class of two person zero-sum linear quadratic differential games, where the control functions for both players are subject to integral constraints. The duration of game is fixed. We obtain the necessary conditions of the Maximum Principle and also optimal control by using method of Pontryagin’s Maximum Principle. Finally, we discuss an example

Optimal control using Pontryagins maximum principle in a linear quadratic differential game.

This paper deals with a class of two person zero-sum linear qu adratic differential games, where the control functions for both players subject to integral constraints. Also the necessary conditions of the Maximum Principle are studied. Main objective in this work is to obtain optimal control by using method of Pontryagin’s Maximum Principle. This method for a time-varying linear quadratic differential game is described. Finally, we discuss about an example

Solution of a linear pursuit-evasion differential game with closed and convex terminal set.

A linear two-person zero-sum pursuit-evasion differential game is considered. Control functions of players are subject to integral constraints. Terminal set is a closed convex subset of The Pursuer tries to bring the state of the system to the terminal set and the Evader prevents bringing of the state to the terminal set where control resource of the Pursuer is greater than that of Evader. We obtain a formula for the optimal pursuit time and construct optimal strategies of the players in explicit form

Solution of a linear pursuit-evasion game with integral constraints.

A linear two player zero-sum pursuit-evasion differential game is considered. The control functions of players are subject to integral constraints. In the game, the first player, the Pursuer, tries to force the state of the system towards the origin, while the aim of the second player, the Evader, is the opposite. We construct the optimal strategies of the players when the control resource of the Pursuer is greater than that of the Evader. The case where the control resources of the Pursuer are less than or equal to that of the Evader is studied to prove the main theorem. For this case a new method for solving of the evasion problem is proposed. We assume that the instantaneous control employed by the Evader is known to the Pursuer. For construction, the strategy of the Evader information about the state of the system and the control resources of the players is used

Linear pursuit-evasion differential games with integral constraints on control functions.

Recently use of decision-making in modern life has extensively increased. This lead to review subject of Pursuit-Evasion (PE) di®erential games. A di®erential game models a situation where two or more players operate in a same environment with con°icting goals. In this work, we attempt to solve general linear PE games in time-varying systems with continuous time. Most studies related to PE games in the current literature concentrate on two-player games with a single Pursuer and a single Evader and the results for general multi-player PE games are still largely sparse. The purpose of this study is to provide a theoretical foundation for linear PE games with integral constraints under the theory of the di®erential game and optimal control theory. The results of this study contain of four parts, in the ¯rst part, the linear pursuit-evasion game by using optimal control techniques which is based on structured controls of the players has been solved. We obtain a formula for the optimal pursuit time and construct the optimal strategies for the players when the control resource of the Pursuer is greater than the Evader. In addition, a new method for solving of the evasion problem is proposed where the control resources of the Pursuer are less than or equal to the Evader. Secondly, the more general linear pursuit-evasion game in the case where the ter-minal set closed and convex has been solved. For this case, we construct the set of attainability which is an ellipsoid. Some conditions on capturability are also discussed. The construction of the optimal pursuit time and optimal strategies for the players are the main objectives of this part. The third part deals with the study of di®erential game of optimal approach with many Pursuers and one Evader, which can be considered as the generalized case of a pursuit-evasion game with one Pursuer and one Evader. This part is devoted to the problem of capture of one Evader by many Pursuers. The case of integral constrains is considered and the strategies for the players are constructed. Con- ditions are obtained for the existence of solutions for a multi-Pursuer game. In order to estimate the value of the game, we obtain several lemmas and theorems. In the fourth part, the optimal control is obtained by using the method of the maximum principle of Pontryagin, where only a special case is studied. The result shows an applications of Pontryagin's maximum principle in a linear quadratic di®erential game (LQDG) with integral constraints

Determine the Optimal Number of Item Groups in the Werehouse Based on ABC Analysis within the Framework of a Supply Chain Network

One of the important techniques in the field of inventory management is inventory classification. ABC analysis is a well-known method which sets the items in a different class, according to their importance and values. In this paper, a three-level supply chain network for a bio-objective optimization model is proposed to improve the quality of inventory grouping based on ABC technique. The first objective maximizes the net profit of the items in the central stock (level 2) and the second objective function maximizes the net profit of items in different wards (level 3) in uncertain condition. The proposed model simultaneously optimizes the service level, the number of inventory groups, and the assigned items to each group based on the set of constraints, inventory shortages, ordering cost, and purchasing cost. To solve this model, the heuristic and exact methods such as LP-metric and modified ε-constraint are applied. Then for comparing those methods, the statistical hypothesis test is used. In conjunction with, the AHP technique is exerted to choose the most efficient solving method. The results show that ε-constraint has the best performance. Ultimately, the proposed model has been implemented in one set of numerical examples to show its applicability in practice

Solution of a linear pursuit-evasion game with integral constraints

A linear two player zero-sum pursuit-evasion differential game is considered. The control functions of players are subject to integral constraints. In the game, the first player, the Pursuer, tries to force the state of the system towards the origin, while the aim of the second player, the Evader, is the opposite. We construct the optimal strategies of the players when the control resource of the Pursuer is greater than that of the Evader. The case where the control resources of the Pursuer are less than or equal to that of the Evader is studied to prove the main theorem. For this case a new method for solving of the evasion problem is proposed. We assume that the instantaneous control employed by the Evader is known to the Pursuer. For construction, the strategy of the Evader information about the state of the system and the control resources of the players is used. References R. Isaacs. Differential games. John Wiley and Sons, New York, 1965. L. S. Pontryagin. Collected works. Nauka, Moscow, 1988. (Russian) L. D. Berkovitz. Necessary conditions for optimal strategies in a class of differential games and control problems. SIAM Journal on Control, 5, 1--24, 1967. L. D. Berkovitz. A survey of differential games. Mathematical Theory of Control, Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, 373--385, 1967. N. N. Krasovskii and A. I. Subbotin. Game-theoretical control problems. New York, Springer, 1988. W. H. Fleming. The convergence problem for differential games. Journal of Mathematical Analysis and Applications. 3, 102--116, 1961. W. H. Fleming. The convergence problem for differential games, Part 2. Advances in Game Theory, Annals of Mathematics Studies, (52), Princeton University Press, Princeton, New Jersey,195--210, 1964. A. Friedman. Differential games. Wiley-Interscience, New York, 1971. R. J. Elliott and N. J. Kalton. The existence of value in differential games. Memoirs of the American Mathematical Society, 126, 1--67, 1972. L. A. Petrosyan. Differential games of pursuit. World Scientific, Singapore, London, 1993. O. Hajek. Pursuit games. Academic Press, New York, San Francisco, 1975. A. Ya. Azimov. Linear differential pursuit game with integral constraints on the control. Differentsial'nye Uravneniya, 11(10), 1975, 1723--1731; English transl. in Differential Equations 11, 1283--1289, 1975. A. Ya. Azimov. A linear differential evasion game with integral constraints on the controls. USSR Computational Mathematics and Mathematical Physics, 14 (6), 56--65, 1974. M. S. Nikolskii. The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR, (2), 49--59, 1969. A. I. Subbotin and V. N. Ushakov. Alternative for an encounter-evasion differential game with integral constraints on the playersí controls. PMM 39(3), 387--396, 1975. V. N. Ushakov. Extremal strategies in differential games with integral constraints. PMM, 36(1), 15--23, 1972. B. N. Pshenichnii and Yu. N. Onopchuk. Linear differential games with integral constraints. Izvestige Akademii Nauk SSSR, Tekhnicheskaya Kibernetika, (1), 13--22, 1968. A. A. Azamov, B. Samatov. $\pi$-strategy. An elementary introduction to the theory of differential games. National University of Uzbekistan. Tashkent, Uzbekistan, 2000. G. I. Ibragimov. A game problem on a closed convex set. Siberian Advances in Mathematics. 12(3), 16--31, 2002. G. I. Ibragimov. A problem of optimal pursuit in systems with distributed parameters. J. Appl. Math. Mech, 66(5), 719--724, 2003. E. B. Lee and and L. Markus. Foundations of optimal control theory, John Wiley and Sons Inc., New York, 1967