Abdulla Azamov

National University of Uzbekistan · Department of Applied Mathematics

In this paper, we consider a third-order explicit scheme based on Taylor's formula to obtain an approximate solution for the Cauchy problem of systems of ODEs. We prove an estimate for the accuracy of the approximate solution with an explicit constant that depends only on the right-hand side of the equation and the domain of the solution.

The purpose of this work is to study the pursuit-evasion problem and the “Life-line” game for two objects (called Pursuer and Evader) with simple harmonic motion dynamics of the same type in the Euclidean space. In this case, the objects move by controlled acceleration vectors. The controls of the objects are subject to geometrical constraints. In...

In this paper, the Pfaff equations with continuous coefficients are considered. Analogs of Peano’s existence theorem and Kamke’s theorem on the uniqueness of the solution to the Cauchy problem are established, and a method for approximate solution of the Cauchy problem for the Pfaff equation is proposed.

Pfaff equations with continuous coefficients are considered. A specific Cauchy problem for a Pfaff equation is transformed to an equivalent system of integral equations of a special type, which is overdetermined. It is shown that in the case of smooth coefficients the consistency of the system is equivalent to the Frobenius integrability criterion....

In this work, the null controllability problem for a linear system in ℓ ² is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in \mathbb {R}$ λ ∈ ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤− 1, which shows the fine difference...

In this paper, the Pfaff equations with continuous coefficients are considered. Analogs of Peanos existence theorem and Kamkes theorem on the uniqueness of the solution to the Cauchy problem are established, and a method for the approximate solution of the Cauchy problem for the Pfaff equation is proposed.

In this work, the null controllability problem for a linear system in $\ell^2$ is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda\in \mathbb R$ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if $\lambda\le-1$, which shows the fine differe...

We study a differential game of many pursuers and one evader. All the players move only along the one-skeleton graph of an orthoplex of dimension d+1. It is assumed that the maximal speeds of the pursuers are less than the speed of the evader. By definition, the pursuit is completed if the position of a pursuer coincides with the position of the ev...

In this paper, a differential game of kind of several pursuers and one evader is studied. All the players move only along the edges of a simplex of dimension [Formula: see text]. The maximal speed of each pursuer is less than that of the evader. If the state of a pursuer coincides with the state of the evader, then pursuit is completed. An exact ma...

We study a differential game of kind of several pursuit points and one evasion point moving along the edges of a regular simplex of dimension d. It is assumed that maximum magnitude of velocity of evader is twice as much as the maximum magnitudes of velocities of pursuers. An exact mathematical formulation of the problem is given by introducing spe...

We consider the differential game between several pursuing points and one evading point moving along the graph of edges of a simplex when maximal quantities of velocities are given. The normalization of the game in the sense of J. von Neumann including the description of classes of admissible strategies is exposed. In the present part of the paper...

In the paper, a four-dimensional model of cyclic reactions of the type Prigogine's Brusselator is considered. It is shown that the corresponding dynamical system does not have a closed trajectory in the positive orthant that will make it inadequate with the main property of chemical reactions of Brusselator type. Therefore, a new modified Brusselat...

This paper is focused on studies of properties of unbounded ω-limit sets of dynamical systems. It is proved that if the ω-limit set Ω is not connected, then each of its components is unbounded, which clarifies the well-known property confirming that if the ω-limit set is not connected then it is unbounded. It is established that the family of conne...

The time-optimal problem for the controllable equation of heat conductivity in a rod is considered. By means of the Fourier expansion, the problem reduced to a countable system of one-dimensional control systems with a combined constraint joining control parameters in one relation. In order to improve the time of a suboptimal control constructed by...

A discrete numerical tracking of the trajectories of dynamic systems is applied to demonstrate the existence of the closed trajectory in a 3D model of cyclic reactions that are associated with a Prigozhin brusselator.

Part II of the paper considers a game between a group of n pursuers and one evader that move along the 1-Skeleton graph M of regular polyhedrons of three types in the spaces ℝd, d ≥ 3. Like in Part I, the goal is to find an integer N(M) with the following property: if n ≥ N(M), then the group of pursuers wins the game; if n < N(M), the evader wins....

This article presents the theoretical framework to solve inverse problems for Delay Differential Equations (DDEs). Given a parameterized DDE and experimental data, we estimate the parameters appearing in the model, using least squares approach. Some issues associated with the inverse problem, such as nonlinearity and discontinuities which make the...

It is discussing one-step methods of numerical solving of a Cauchy problem for systems of ordinary differential equations. It is constructed an algorithm of numerical solving with arbitrary high precision for quadratic systems based on a context-free grammar of N. Chomsky, that generates a special language of terms of Taylor’s formula. The estimati...

In this paper the effectiveness and applicability of the DN-tracking method for constructing Poincaré maps are investigated. The DN-tracking method is demonstrated by samples of dynamical systems having closed trajectories, bifurcations of homoclinical loop of a saddle and period doubling.

A simplified mathematical model of a rotary regenerative air pre–heater (RRAP) is suggested and studied based on the averaged dynamics of the heat exchange process between nozzles and a heat carrier (i.e. air or gas–smoke mixture). Averaging in both spatial coordinates and time gives a linear discrete system that allows deriving explicit formulas f...

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’...

Common used Poincaré’s method to study of qualitative behavior of polynomial systems at infinity is revisited. An effect called “loss of projectivity” in dynamical systems of even degree is accented. It is given an interpretation of Lefschetz’s equation in terms of Pfaff forms and a modified Lefschetz’s equation is suggested which allows to get pro...

Abstract: We consider a game between a group of n pursuers and one evader moving with the same maximal speed along 1-skeleton of a given regular polyhedron. The objective of the paper consists of finding an integer N(M) possessing the following property: if n>N(M) then the group of pursuers wins while if n < N(M) then an evader wins. Part I of the...

A simple motion evasion differential game of two evaders and one pursuer was studied. Control functions of all players are subjected to integral constraints. We say that evasion is possible if the state of at least one of the evaders does not coincide with that of the pursuer. We proved that if the total energy of the evaders is greater than or equ...

Structure of the phase space of the nonlinear system ${{\dot {\varvec x} = {\varvec Ax} + {\langle {\varvec a}, {\varvec x}\rangle {\varvec x}}}}$ is clarified using saddle-node bifurcations ${{{\varvec x}, {\varvec a} \in \mathbb{R}^d,}}$ is a d × d-matrix).

Existence of a limit cycle for the dynamical system well-known as the Prigogine brusselator model is proved when parameters of the system take concrete values. The proof is conducted by the new method called discrete numerical (DN) tracking of trajectory combined with the Poincare-Bendickson theorem.

We describe an example of a three-dimensional linear differential game with convex compact sets of control. In this example, the integrand in Pontryagin’s first direct method is discontinuous on a set of positive measure. Keywordslinear differential game–Pontryagin direct method–convex compact set of control–stroboscopic strategy–Lebesgue measurab...

The DN-tracking method is used to prove the existence of a closed trajectory in a quadratic system of ordinary differential equations in three dimensions. Keywordsdynamical system–closed trajectory–Poincaré map–numerical methods

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 strategie...

Linear discrete controlled systems and linear discrete pursuit games are considered in the problem where the control vector of the pursuer is subject to a geometrical constraint and the control of the pursued object is subject an integral constraint. Necessary and sufficient conditions of solvability of the 0-controllability problem and that of com...

It is considered the problem connected with the combinatorial metric approach to the notion of solution of matrix games. According to this approach it is searched a matrix B that possesses equilibrium and is the closest to the given matrix A in the sense of some metric d(A,B). In the case when d(A,B) is the number of pairs (i,j) such that a ij ≠b i...

We obtain a lower bound for the pursuing (searching) point velocity for which the minimax dynamic graph search problem in N.N. Petrov’s setting is solvable.

A relation is established between problems of pursuit, controllability and stability in the large in linear systems when a geometric constraint is imposed on the control vector of the pursuer and an integral constraint is imposed on the control function of the evader.

A solution to the pursuit problem for one linear differential game, critical in the sense that it lies on the boundary of solvability of the approach and evasion problems, is given. The result thus obtained is used to answer two question connected with Pontryagin’s methods.

The structure of phase space of differential pursuit-evasion games is studied for the case when the evader is subjected to information discrimination with an advance by δ > 0, δ = const. The method of transf inite iteration of the Pshenichnyi operator is used to establish an alternative for differential pursuit-evasion games in an infinite time int...

An analogue of the Pontryagin alternating integral is introduced; a study is made of its properties, of the connection with the upper Pontryagin integral, and of an application to a pursuit problem; and a duality of theoretical significance in the Pontryagin theory is established.Bibliography: 12 titles.

A differential game is considered, in which a pursuer can move over the whole plane with unit velocity. while the escaper can move along a prescribed curve with a bounded velocity greater than unity. An escape strategy is constructed, ensuring a positive constant lower bound for the distance between the players.

Optimal strategies (that is, strategies constituting an equilibrium situation) in a game between two persons with zero sum in which each of the players chooses a point from compact convex subsets of a Euclidean space and the payment is the Euclidean distance between the points chosen are constructed.

Proof. Suppose x(t) is a solution such that x(0) = 0, x(0) = 1. Then X(x) = a > 0. Indeed, it follows from assumption A that each solution of (t) has at most one zero. Consequently, x(t) > 0 for t > 0. This implies that x(t) increases monotonically, hence its exponent cannot be equal to -a. Suppose y(t) is a solution satisfying the initial conditio...

The notion of the strategy of parallel pursuit (briefly Π-strategy) was introduced and used to solve the quality problem in the “game with a survival zone” in [L. A. Petrosyan, Differentsial’nye igry preslodovaniya (Russian). Leningrad: Izdatel’stvo Leningradskogo Universiteta (1977; Zbl 0457.90087)]. Later on, there were found other applications o...