Ivan Polekhin

Ivan Polekhin
Russian Academy of Sciences | RAS · Steklov Mathematical Institute

PhD

About

24
Publications
1,397
Reads
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94
Citations
Education
September 2003 - June 2008
Lomonosov Moscow State University
Field of study
  • Mechanics and Mathematics

Publications

Publications (24)
Preprint
In the paper we consider systems in oscillating force fields such that the classical method of averaging can be applied. We present sufficient conditions for the existence of forced oscillations in such systems and study the asymptotic behaviour of some solutions. In particular, we show that for an inverted pendulum with a horizontally moving pivot...
Preprint
In this short paper we consider a possible application of the Wa\.zewski topological method to feedback control systems. We show how this method can be efficiently used to prove the impossibility of global stabilization in such problems.
Preprint
In the paper we study the existence of a forced oscillation in two Lagrange systems with gyroscopic forces: a spherical pendulum in a magnetic field and a point on a rotating closed convex surface. We show how it is possible to prove the existence of forced oscillations in these systems provided the systems move in the presence of viscous friction.
Preprint
We present sufficient conditions for the existence of forced oscillations in non-autonomous mechanical systems. Previously, similar results were obtained for systems with friction. Presented results hold both for systems with and without friction. Some examples are given.
Article
The change of the precession angle is studied analytically and numerically for two classical integrable tops: the Kovalevskaya top and the Goryachev — Chaplygin top. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or can be...
Preprint
Full-text available
The change of the precession angle is studied analytically and numerically for two classical integrable tops: the Kovalevskaya top and the Goryachev-Chaplygin top. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or can be es...
Article
The problem of motion of a rigid body with a fixed point is considered. We study qualitatively the solutions of the system after Routh reduction. For the Lagrange integrable case, we show that the trajectories of solutions starting at the boundary of a possible motion area can both cover and not cover the entire possible motion area. It distinguish...
Article
An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizon are presented. The proof is based on...
Article
It is shown that the problem of the asymptotic stabilization of a given position of the Lagrange top for any control from a sufficiently wide class does not permit the existence of a single uniformly asymptotically stable equilibrium, even with possible impacts of the top against the horizontal plane; i.e., the global stabilization of the system is...
Preprint
Full-text available
An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizontal positions are presented. The proof...
Conference Paper
We consider a system of a planar inverted pendulum in a gravitational field. First, we assume that the pivot point of the pendulum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizontal (never falls)...
Article
Full-text available
We consider the problem of transfer of a Lagrangian system with gyroscopic forces to a given state in the configuration space by means of an impulsive isoenergetic control. This type of control can change instantly the direction of the generalized velocity, but preserve the total energy of the system. We present sufficient conditions for global con...
Article
Full-text available
We consider a classical problem of control of an inverted pendulum by means of a horizontal motion of its pivot point. We suppose that the control law can be non-autonomous and non-periodic w.r.t. the position of the pendulum. It is shown that global stabilization of the vertical upward position of the pendulum cannot be obtained for any Lipschitz...
Article
Full-text available
We consider two classical celestial-mechanical systems: the planar restricted circular three-body problem and its simplification, the Hill's problem. Numerical and analytical analyses of the covering of a Hill's region by solutions starting with zero velocity at its boundary are presented. We show that, in all considered cases, there always exists...
Article
We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these...
Article
Full-text available
Consider a Lagrangian system with the Lagrangian containing terms linear in velocity. By analogy with the systems in celestial mechanics, we call a bounded connected component of the possible motion area of such a system a Hill's region. Suppose that the energy level is fixed and the corresponding Hill's region is compact. We present sufficient con...
Article
Full-text available
Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface, possibly with a boundary, in an external periodic field. We present sufficient conditions for the existence o...
Article
Full-text available
We consider the forced motion of a relativistic particle constrained on a curve and present sufficient conditions for periodic oscillations by means of an illustrative geometrical approach. Obtained result is illustrated by a few examples including the forced relativistic pendulum.
Article
Full-text available
We present sufficient conditions for the existence of a periodic solution for a class of systems describing the periodically forced motion of a massive point on a compact surface with a boundary.
Article
Full-text available
For the system of an inverted spherical pendulum with friction and a periodically moving pivot point we prove the existence of at least one periodic solution with the additional property of being falling-free. The last means that the pendulum never becomes horizontal along the considered periodic solution. Presented proof is an application of some...
Article
Full-text available
Two examples concerning an application of topology in the study of the dynamics of an inverted plain mathematical pendulum with a pivot point moving along a horizontal straight line are considered. The first example is an application of the Wa{\.z}ewski principle to the problem of the existence of a solution without falling in the case of a arbitra...

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