On transfer problems in control systems

The model of the considered system is described by partial differential equations. The players’ (opponents’) control parameters occur on the right-hand side of the equation and are subjected to various constraints (geometric and integral). The first player’s goal is to bring the system from one state into another desired state; the second player’s goal is to prevent this from happening. We represent new sufficient conditions for bringing the system from one state into another (in particular, into the origin).     

On the theory of three-person differential games

A differential game of three players with dynamics described by linear differential equations under geometric constraints on the control parameters is considered. Sufficient conditions are obtained for the existence of the first player’s strategy guaranteeing that the trajectory of the game reaches a given target set for any admissible control of the second player and avoids the terminal set of the third player. An algorithm of constructing the first player’s strategy guaranteeing the game’s termination in finite time is suggested. A solution of a model example is given.     

On pursuit problems in controlled distributed systems

In the present paper, we study the problem of transferring a system from one state into another state in the presence of a player trying to prevent the occurrence of this transfer in the system. The system dynamics is described by partial differential equations whose right-hand sides contain the player controls in additive form. In a similar setting, the problem was solved in several papers, but it has not been considered for the case in which various constraints are imposed on the controls of the players. Here, in contrast to several other papers, we consider games in the entire scale of spaces H _r , r ≥ 0. We propose a new approach for completing the pursuit under various constraints on the player controls.     

On a noisy-silent-versus-silent duel with equal accuracy functions

This paper deals with the noisy-silent-versus-silent duel with equal accuracy functions. Player I has a gun with two bullets and player II has a gun with one bullet. The first bullet of player I is noisy, the second bullet of player I is silent, and the bullet of player II is silent. Each player can fire their bullets at any time in [0, 1] aiming at his opponent. The accuracy function ist for both players. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is –1. The optimal strategies and the value of the game are obtained. Although optimal strategies in past works concerning games of timing does not depend on the firing moments of the players, the optimal strategy obtained for player II depends explicitly on the firing moment of player I's noisy bullet.     

On the noisy-silent versus silent-noisy duel with equal accuracy functions

This paper deals with the noisy-silent versus silent-noisy duel with equal accuracy functions. Each of player I and player II has a gun with two bullets and he can fire his bullets at any time in [0, 1] aiming at his opponent. The first bullet of player I and the second bullet of player II are noisy, and the second bullet of player I and the first bullet of player II are silent. It is assumed that both players have equal accuracy functions. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is ?1. The value of the game and the optimal strategies are obtained. We find that the firing time of the silent bullet by player II's optimal strategy depends directly on the firing time of player I's noisy bullet.     

Stochastic Discrete-Time Nash Games with Constrained State Estimators

In this paper, we consider stochastic linear-quadratic discrete-time Nash games in which two players have access only to noise-corrupted output measurements. We assume that each player is constrained to use a linear Kalman filter-like state estimator to implement his optimal strategies. Two information structures available to the players in their state estimators are investigated. The first has access to one-step delayed output and a one-step delayed control input of the player. The second has access to the current output and a one-step delayed control input of the player. In both cases, statistics of the process and statistics of the measurements of each player are known to both players. A simple example of a two-zone energy trading system is considered to illustrate the developed Nash strategies. In this example, the Nash strategies are calculated for the two cases of unlimited and limited transmission capacity constraints.     

Control design in problems with an unknown level of dynamic disturbance

Linear differential games for two players with a fixed termination time are considered. The objective of player 1 is to bring the motion of the system into an assigned terminal set or fairly close to it at the termination time. Player 2 (the disturbance) opposes this. The control of player 1 is scalar and its absolute value is subject to a constraint. One special feature of the formulation is that no constraint on the control of player 2 is specified a priori. A method of designing the feedback control of player 1 that works satisfactorily over a broad range of disturbance levels and corresponds to a small magnitude of the control input at a low level of disturbance is proposed and verified. A numerical program is written for the case of small dimensionality of the phase variable. The results of the simulation of a system that describes a conflict-controlled pendulum are presented.     

On finite tetraprimary groups

A three-player game is considered in which the first and second players have dynamic superiority over the third player. Two fixed time points are specified. The game ends if either the first player captures the third player at the first time point, or the second player captures the third player at the second time point. We analyze a situation when the initial positions in the game are such that neither the first nor the second player alone can capture the third player at the specified points of time. We propose sufficient conditions on the parameters of the game under which, for given initial states of the players, the first and second players by applying some controls can guarantee that one of them will meet the third player at the prescribed moment. Simulation results for a model example are also presented.     

Control of an aircraft landing in windshear

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis.The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite.The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations.The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.     

The construction of stable bridges in differential games with phase constraints

A differential approach-and-evasion game in a finite time interval is considered [1]. It is assumed that the positions of the game are constricted by certain constraints which represent a closed set in the space of the positions. In the case of the first player, it is necessary to ensure that the phase point falls into the terminal set at a finite instant of time and, in the case of the second player, that this terminal set is evaded at this instant [1]. A method is proposed for the approximate construction of the positional absorption set, that is, the set of all positions belonging to a constraint from which the problem of approach facing the first player is solvable. Relations are written out which determine the system of sets which approximates the positional absorption set. The main result is a proof of the convergence of the approximate system of sets to the positional absorption set and the construction of a computational procedure for constructing the approximate system of sets.     

Rendezvous search on the line with bounded resources: expected time minimization

Two players are placed on the line and want to meet. Neither knows the direction of the other, but they know the distance between them or perhaps the distribution of this distance. They can move with speed at most one, and each has a ‘resource constraint’ on the total distance he can travel. We first consider the question of whether the two players can ensure that they meet. When they can, then we seek the least expected meeting time. Otherwise, we maximize the probability of a meeting.This generalizes the similar problem studied by the first author and S. Gal [SIAM J. Control and Optimization 33/4 (1995) 1270] without any resource constraint, and indeed for sufficiently large resources gives the same rendezvous time, i.e. least expected time to meet. The paper may also be considered a generalization of the Linear Search Problem, studied by the second author and others, which corresponds to the case when one of the resource constraints is zero, so that player cannot move. More specifically, it generlaizes the bounded resource version of the Linear Search Problem presented by Foley, Hill and Spruill [Naval Research Logistics 38 (1991) 555–565].     

Estimating the winning chances of a solution suggested by the first player under conditions of voting by veto

This work studies voting according to the veto model used by a board of directors to choose between several ways of development of a corporation. Several rules for choosing are considered, including the possibility of a certain player determining the order of the other players’ moves. The specified task is to find the most advantageous preferences of the voters from the viewpoint of the player who makes his choice first. Of special interest is a situation in which one of the players predominates and can determine the order of the other voters’ moves. The problem is solved for three players both in a case of strict preferences and in a case where the voters decide between several equally preferable ways of developing the corporation.     

On the Approximate Controllability of Stackelberg–Nash Strategies for Linearized Micropolar Fluids

We study a Stackelberg strategy subject to the evolutionary linearized micropolar fluids equations, considering a Nash multi-objective equilibrium (non necessarily cooperative) for the “follower players” (as is called in the economy field) and an optimal problem for the leader player with approximate controllability objective. We will obtain the following three main results: the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability of the linearized micropolar system with respect to the leader control, and the existence and uniqueness of the Stackelberg–Nash problem, where the optimality system for the leader is given.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

On the weak control of slowly damped systems : PMM vol. 42, no. 6, 1978. pp. 1000–1005

Asymptotic formulas are obtained which make it possible to derive the first approximation solution of the Riccati matrix algebraic equation of special form. Method is based on Bass' formulas [1] and the theory of perturbations [2], The problem of control of a slowly damped oscillator is investigated in detail. Formulation of the problem in this paper differs substantially from that in [3] (no assumption is made about single-frequency oscillations, and only a stationary system is considered over an infinite time interval).     

Synthesis of a control in a non-linear dynamical system based on decomposition

A non-linear controllable dynamical system with many degrees of freedom, described by Lagrange equations of the second kind, is considered. Geometric constraints are imposed on the magnitudes of the controls. It is assumed that, in the equations of motion, the kinetic energy matrix is close to a certain constant diagonal matrix. It is possible, for example, to reduce the equations of motion of robots, the drives of which have large gear ratios, to a system of this kind. A problem is formulated on the transfer of a system in a finite time from a specified initial state to a final state with zero velocities. The method of decomposition [1] is used to construct the equations. Sufficient conditions are found subject to which the maximum values of the non-linear terms in the equations of motion do not exceed the permissible magnitudes of the controls. In this case, non-linearities are treated as limited perturbations and the system is decomposed into independent, linear, second-order subsystems. A feedback control is specified for these subsystems which guarantees that each of them is brought into the final state for any permissible perturbations. The control has a simple structure. Applications of the proposed approach to problems in the control of manipulating robots are considered.     

Small-parameter method for constructing approximate strategies in a class of differential games : PMM vol. 39, n 6, 1975, pp. 1006–1016

We examine a class of problems in which the pay-off is some function of the terminal state of a conflict-controlled system. When the opportunities of one of the players are small in relation with the opportunities of the other, we propose methods for constructing approximate optimal strategies of the players, based on solving the Bellman equation containing a small parameter. We have shown that the players' approximate optimal strategies can be constructed if the solutions of the corresponding optimal control problems are known. The error bounds for the methods are proved and examples are considered. The arguments used rely on the results in [1–6] on the theory of differential games and on [7–11] devoted to optimal control synthesis methods for systems subject to random perturbations of small intensity.     

On the concepts of rationalizability in games

Rationalizability arises when the decision situations and rational behaviors of the players are common knowledge among them. We extend the notion of rationalizability, introduced by Bernheim [5] and Pearce [18] for Bayesian behavior, to some another kinds of player's behavior. We also present a representation of common knowledge consisting in introducing an additional player who sends messages to the players. This revised version was published online in June 2006 with corrections to the Cover Date.     

An alternative in the differential-difference game of approach — evasion with a functional target : PMM vol. 40, n 6, 1976, pp.987–994

An approach-evasion problem with a functional target set under constraints on the system's trajectory is studied for a conflict-con trolled system described by a differential-difference equation. The main result states: either a strategy exists for the first player resolving the approach problem or a strategy exists for the second player resolving the evasion problem. The paper is closely related to [1–6].