Game Problem Of Approach Research Articles

Direct Method for Solving Game Problems of Approach of Controlled Objects

Direct Method for Solving Game Problems of Approach of Controlled Objects

Modified Resolving-Functions Method for Game Problems of Approach of Controlled Objects with Different Inertia

Modified Resolving-Functions Method for Game Problems of Approach of Controlled Objects with Different Inertia

Transfinite Version of the Program Iteration Method in a Game Problem of Approach for an Abstract Dynamical System

Transfinite Version of the Program Iteration Method in a Game Problem of Approach for an Abstract Dynamical System

On the Relaxation of a Game Problem of Approach with Priority Elements

On the Relaxation of a Game Problem of Approach with Priority Elements

Method of Resolving Functions for Game Problems of Approach of Controlled Objects with Different Inertia

The authors consider the problem of approach of controlled objects with different inertia in dynamic game problems on the basis of the modern version of the method of resolving functions. For such objects, it is characteristic that the Pontryagin condition is not satisfied on a certain time interval, which significantly complicates the application of the method of resolving functions to this class of dynamic game problems. A method for solving such problems is proposed, which is associated with the construction of some scalar (resolving) functions, which qualitatively characterize the course of approach of controlled objects with different inertia and the efficiency of the decisions made. The method of resolving functions allows efficient use of the modern technique of multi-valued mappings in substantiating game constructions and obtaining meaningful results on their basis. The guaranteed times of game termintion are compared for different schemes of approaching of controlled objects. An illustrative example is given.

On a Differential Game in a Stochastic System

We study the game problem of approach for a system whose dynamics is described by a stochastic differential equation in a Hilbert space. The main assumption on the equation is that the operator multiplying the system state generates a strongly continuous semigroup (a semigroup of class $$C_{0}$$ ). Solutions of the equation are represented by a stochastic variation of constants formula. Using constraints on the support functionals of sets defined by the behavior of the pursuer and the evader, we obtain conditions for the approach of the system state to a cylindrical terminal set. The results are illustrated with a model example of a simple motion in a Hilbert space with random perturbations. Applications to distributed systems described by stochastic partial differential equations are considered. By taking into account a random external influence, we consider the heat propagation process with controlled distributed heat sources and sinks.

Game Problems of Approach for Quasilinear Systems of General Form

A conflict-controlled process of the approach of a trajectory to a cylindrical terminal set is studied. The problem statement encompasses a wide range of quasilinear functional-differential systems. We use the technique of set-valued mappings and their selections to derive sufficient conditions for the game termination in a finite time. The methodology used is close to the scheme that involves the time of the first absorption. By way of illustration, quasilinear integro-differential games are examined. For this purpose, their solutions are presented in the form of an analog of the Cauchy formula. The calculations are performed for the case of a system with a simple matrix; the control sets of the players are balls centered at the origin and the terminal set is a linear subspace. Depending on the relations between the initial state of the system and the parameters of the process, sufficient conditions for the game termination are derived. An explicit form of the guaranteed time is found in one specific case.

An Addition to the Definition of a Stable Bridge and an Approximating System of Sets in Differential Games

We present some modifications of the definitions of a u-stable bridge and of an approximating system of sets in a game problem of approach at a fixed instant of time. These modifications are aimed at developing and substantiating algorithms for approximate construction of solutions in approach game problems.

On a differential game in an abstract parabolic system

We consider the game problem of approach for a system whose dynamics is described by a differential operator equation in a Hilbert space. The equation is written in an implicit form with generally noninvertible operator multiplying the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in a right half-plane. Using the method of resolving functionals, we obtain sufficient conditions for the approach of a dynamical vector of the system to a cylindrical terminal set. Applications to systems described by partial differential equations are considered.

On a differential game in a system with distributed parameters

We consider a game problem of approach for a system whose dynamics is described by a partial differential equation not of Kovalevskaya type, i.e., unsolved with respect to the time derivative. The equation with boundary conditions is written in a Hilbert function space in an abstract form as a differential operator equation. Using the method of resolving functionals, we obtain sufficient conditions for the approach of the system’s dynamical vector to a cylindrical terminal set. The results are exemplified by means of a model problem concerning a filtering process for a fluid in fractured porous rocks.

An estimate of the weak invariance defect of sets with piecewise smooth boundary

We consider a nonlinear control system and a differential inclusion corresponding to it on a finite time interval. We study a game problem of approach of the control system to a compact target set at a fixed time. We study some questions related to the weak invariance of sets with respect to the differential inclusion and related to calculating and estimating the weak invariance defect of sets with piecewise smooth boundaries. Under conditions that provide the correctness of the notion of weak invariance defect, we derive formulas for calculating the weak invariance defect of a set with piecewise smooth boundary of quite general form. We consider a model two-dimensional control system.

On the question of the stability defect of sets in an approach game problem

The stability property is studied in a game problem of approach of a conflictcontrol system to a target set at a fixed termination time. The notion of the stability defect of sets in the space of game positions is investigated.

On a supplement to the stability property in differential games

The stability property in a game problem of approach “at the final instant” is studied. The concept of the stability defect of sets in the position space of a game is analyzed.

On solving a game problem of approach at a fixed time

A game problem of approach at a fixed time facing the first player is considered. An approximate method is suggested for constructing the positional absorption set, which is the solvability set for the approach problem. Recurrence relations are written that define a system of sets approximating the positional absorption set in the phase space of the conflict-control system. A resolving procedure of control with a guide is described, which copies the controls.

Coincidence criteria for maximal stable bridges in two game problems of approach

We consider a finite-dimensional conflict-controlled system whose behavior on a finite time interval is described by a vector differential equation. We analyze two game problems of approach in the phase space. In both problems the same terminal set is considered: in the first case, one should guarantee that the phase vector of the system reaches the terminal set at the final instant of time; in the second case, the phase vector should reach the terminal set no later than the final time instant. It is natural to assume that the construction of a solution to the first problem is much simpler than the construction of a solution to the second problem; this fact is confirmed by available experience. The paper is devoted to finding conditions on the system and the terminal set under which the solutions to the above problems coincide. Using these conditions, one can replace the solution of the second problem by the simpler solution of the first problem.

Game problems for fractional quasilinear systems

We present a general method for solving game problems of approach for dynamical systems with Volterra evolution. This method is based on the method of resolving functions and uses the apparatus of the theory of set-valued mappings. In more detail, we study game problems for systems with Riemann-Liouville fractional derivatives and regularized derivatives of Dshrbashyan-Nersesyan (fractal games) on the basis of the generalized matrix functions of Mittag-Leffler introduced here.

Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order

A general method for solving game problems of approach for dynamic Volterra-evolution systems is presented. This method is based on the method of resolving functions [5] and the techniques of the theory of multivalued mappings. Properties of resolving functions are studied in more detail. Cases are separated where resolving functions can be derived in an analytical form. The scheme proposed covers a wide range of functional-differential systems, in particular, integral, integro-differential, and differential-difference systems of equations that describe the dynamics of a conflict controlled process. Game problems for systems with fractional Riemann-Liouville derivatives and regularized Dzhrbashyan-Nersesyan derivatives are studied in more detail. We will call them fractal games. An important role in the presentation of solutions of such systems is played by the generalized Mittag-Leffler matrix functions, which are introduced here. The use of asymptotic representations of these functions within the framework of the scheme of the method allows us to establish sufficient conditions of resolvability of game problems. A formal definition of parallel approach is given and illustrated by game problems for systems with fractional derivatives.

Dynamic Game Problems of Approach for Fractional-Order Equations

We propose a general method for the solution of game problems of approach for dynamic systems with Volterra evolution. This method is based on the method of decision functions and uses the apparatus of the theory of set-valued mappings. Game problems for systems with Riemann–Liouville fractional derivatives and regularized Dzhrbashyan–Nersesyan derivatives (fractal games) are studied in more detail on the basis of matrix Mittag-Leffler functions introduced in this paper.

Game problem of approach under failure of controlling devices

The linear differential game of approaching given terminal set by a trajectory of the conflict controlled process is treated here. The process evolves in such a way that at some a priori unknown instant of time the control of the first player is shut off for a certain period of time, which is necessary for the elimination of the breakdown and which duration is known in advance. Then the process lasts up to the moment when its trajectory hits the terminal set. On the basis of technique, leaning upon the resolving function's construction, developed by the author, the sufficient conditions of two kinds for solvability of the approach problem are derived in the paper. The results are illustrated by the model example of the game problem with “simple motions” dynamics.