Journal of Applied Mathematics and Mechanics 73 (2009) 411–420 Contents lists available at ScienceDirect Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech Extremal aiming in problems with an unknown level of dynamic disturbance  S.A. Ganebnyi, S.S. Kumkov, V.S. Patsko ∗ Ekaterinburg, Russia article info Article history: Received 23 December 2008 abstract The method of extremal aiming, well-known in the theory of differential games, is applied to problems in which the level of dynamic disturbance is not stipulated in advance. Problems with linear dynamics, a fixed termination time and a geometric constraint on the effective control are considered. The aim of the control is to bring the system into a specified terminal set at the instant of termination. A feedback control method is proposed which ensures successful completion if the disturbance does not exceed a certain critical level. Here, “weak” disturbance is countered by a “weak” effective control. A guarantee theorem is formulated and proved. An illustrative example is considered. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Methods for solving problems in which geometric constraints on the control actions of both players are stipulated according to the formulation, are well developed in the theory of antagonistic differential games. 1–5 However, in many practical problems, a geometric constraint is only imposed on the effective control (on the control of the first player) while the imposition of such a constraint on the dynamic action of the disturbance (on the control of the second player) is unnatural. Moreover, the optimal feedback control of the first player, obtained within the limits of the standard formalization of an antagonistic differential game is directed to countering the worst disturbance. In real situations, the dynamic disturbance does not, as a rule, act in the worst way. It is desirable to have a feedback control process which operates successfully over a wide range of disturbance. Here, the “weaker” or “less optimal” the disturbance, the “weaker” the effective control countering it must be. The aim of this paper is to propose such a method which rests on the established theory of differential games. The problem considered is similar to problems of suppressing a bounded external disturbance by a control system, which are being intensively studied at present. 6–9 The main difference in this paper, apart from the mathematical apparatus used, lies in the fact that the control process in it is considered in a finite time interval and the effective control is constrained by a geometric limitation, according to the formulation of the problem. Among the papers, that use the results of the theory of differential games and are orientated towards problems with an unknown level of disturbance, we mention Ref. 10. The central concept used in this paper is the concept of a stable bridge. 3–5 A set in a time × phase vector space, in which the first player, by using his control and discriminating the adversary, can maintain the motion of the system right up to the instant of termination, is called a stable bridge. Consider a family of differential games where the geometric constraint on the second player’s control depends on a scalar parameter. We will associate a certain constraint on the first player’s control and a certain stable bridge with each value of the parameter. We will assume that the family of bridges is arranged in the order of increasing values of this parameter. The first player guarantees the retention of the phase vector in the tube of a stable bridge using his control, the level of which corresponds to the tube being considered if the second player’s control also satisfies the corresponding constraint. The family of bridges enables us to construct the first player’s feedback control and to describe the guarantee ensured by the control. We will now explain how this takes place. Suppose a disturbance, which does not exceed a certain level, acts on a control system. The motion of the system will then intersect bridges of the family which has been constructed until the boundary of the bridge is reached (from above or below) corresponding to the level of disturbance realized. The motion will subsequently proceed within the limits of this bridge. Fine tuning (adaptation) of the level of the effective control therefore occurs automatically under a level of disturbance, unknown in advance.  Prikl. Mat. Mekh. Vol. 73, No. 4, pp. 573 - 586, 2009. ∗ Corresponding author. E-mail address: patsko@imm.uran.ru (V.S. Patsko). 0021-8928/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2009.08.010