Multivalued mappings and their selectors in the theory of conflict-controlled processes

Чикрий, А. А.; Rappoport, I. S.; Chikrii, K. A.

doi:10.1007/s10559-007-0097-8

Cited by 17 publications

(3 citation statements)

References 9 publications

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“…In each case, we have obtained sufficient solvability conditions for the approach problem in a finite time. The paper continues the studies started in [19] and employs the results from [9].…”

Section: Introductionmentioning

confidence: 91%

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Game approach problems for dynamic processes with impulse controls

Химич

Chikrii

2009

Cybern Syst Anal

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Section: Introductionmentioning

confidence: 91%

“…(11), let us add and subtract Let now x g ( , , ) (5) and (6) and Condition 1. Accordingly, a ( ) T = + ¥ and using (9) to construct the functions a i T ( ) yields an uncertainty. Therefore, we select the saltus vectors u i , u U i Î , i T = 1, .…”

Section: Condition 3 the Multivalued Mappingsmentioning

confidence: 99%

Game approach problems for dynamic processes with impulse controls

Химич

Chikrii

2009

Cybern Syst Anal

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“…These issues are studied, for example, in [27]. Various optimization techniques can be applied in this case.To study discrete systems with random perturbations, as the base method we will use the method of resolving functions [5,28] well known in dynamic game theory. The ability of evaluating resolving functions in analytic form for a rather wide class of problems allows concluding on whether it is possible to bring the trajectory of the process to a prescribed set with account for the probabilistic characteristics of the random perturbation.…”

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An approach problem for a discrete system with random perturbations

Dziubenko

Чикрий

2010

Cybern Syst Anal

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The method of resolving functions is used to prove general sufficient conditions for the finiteness of the guaranteed time of reaching a cylinder terminal set by a quasilinear conflict-controlled process with random perturbations. The conditions of finiteness almost everywhere and finiteness with positive probability for the guaranteed reaching time are found for a process with simple matrix. The theory of dynamic games is developed well enough. The available approaches such as Pontryagin's direct methods [1], Krasovskii's extremal aiming principle [2], Pshenichnyi's method of semigroup operators [3], the technique related to the basic equations of Isaac's differential-game theory [4], the method of resolving functions [5], and other efficient procedures allow testing wide classes of conflict-controlled processes for game approach of trajectories. Evasion methods are various and convincing: the Pontryagin-Mishchenko method of evasion maneuver, the methods of constant and variable directions, the method of invariant subspaces, the recursive method (see [6] for a review).Therefore, there naturally arises an issue of introducing not only deterministic [7] but also stochastic uncertainty into the model of a conflict-controlled process. This can be done in different ways. For example, the studies [8-10] consider the situation where not the initial state of a process but only its distribution function is known, which leads to the Fokker-Planck-Kolmogorov equation [8,10,11] whose solution is the distribution function of the current state of the process. The optimization of such continuous processes are exemplified in [8] (based on the Pontryagin's maximum principle) and in [10], where the Milyutin-Dubovitsky necessary extremum conditions are employed. Since solving the Fokker-Planck-Kolmogorov equation in the continuous case is a challenge, the initial distribution or time are often discretized. The complete pattern of possible formalizations in this class is presented in [11]. Such studies for game formulations are exemplified in [12,13].The above-mentioned problems with stochastic uncertainty are often called search problems [14]. The studies [15][16][17][18] propose a bilinear search model where the stochastic transition matrix is a control unit, the number of states is finite, and the time is discrete. The performance criterion is the detection probability or average time of detection of an object. The discrete maximum principle or dynamic programming allows optimizing the search process, including the participation of groups of moving objects under various information assumptions.Uncertainty can be introduced in the model by mixed strategies of the players [19,20]. Various classes of stochastic games are studied in [21][22][23][24][25][26]. Apparently, one of the most natural ways to study stochastic conflict-controlled processes (stochastic differential games) is to consider the stochastic Ito equation with control in the original formulation or to introduce a random perturbation to the right-hand side of the dete...

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Riemann–Liouville, Caputo, and Sequential Fractional Derivatives in Differential Games

Чикрий

Matychyn

2010

Annals of the International Society of Dynamic Games

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There exists a number of definitions of the fractional order derivative. The classical one is the definition by Riemann-Liouville [1]. The RiemannLiouville fractional derivatives have a singularity at zero. That is why differential equations involving these derivatives require initial conditions of special form lacking clear physical interpretation. These shortcomings do not occur with the regularized fractional derivative in the sense of Caputo. Both the Riemann-Liouville and Caputo derivatives possess neither semigroup nor commutative property. That is why so-called sequential derivatives were introduced by Miller and Ross [2].In this paper we treat sequential derivatives of special form [3]. Their relation to the Riemann-Liouville and Caputo fractional derivatives and to each other is established.Differential games for the systems with the fractional derivatives of Riemann-Liouville, Caputo, as well as with the sequential derivatives are studied. Representations of such systems' solutions involving the MittagLeffler generalized matrix functions [4] are given. The use of asymptotic representations of these functions in the framework of the Method of Resolving Functions [5,6,7] allows to derive sufficient conditions for solvability of corresponding game problems. These conditions are based on the modified Pontrjagin's condition [5]. The results are illustrated on a model example where a dynamic system of order π pursues another system of order e.

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Multivalued mappings and their selectors in the theory of conflict-controlled processes

Cited by 17 publications

References 9 publications

Game approach problems for dynamic processes with impulse controls

Game approach problems for dynamic processes with impulse controls

An approach problem for a discrete system with random perturbations

Riemann–Liouville, Caputo, and Sequential Fractional Derivatives in Differential Games

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