Функциональный метод локализации и принцип инвариантности Ла-Салля

A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space.Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions.Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.В задачах качественного анализа динамических систем хорошо зарекомендовал себя функциональный метод локализации. Предложенный в 90-хх гг., он активно использовался в исследовании ряда известных систем дифференциальных уравнений, как автономных, так и неавтономных, дискретных систем, в том числе систем включающих управление и/или возмущения.Суть метода состоит в построении такого множества в фазовом пространстве динамической системы, которое содержит все инвариантные компактные множества. Понятие инвариантного компактного множества включает положения равновесия, предельные циклы, аттракторы, репеллеры и другие структуры в фазовом пространстве системы, играющие важную роль в описании поведения динамической системы. Построенное множество называют локализирующим. Оно служит внешней оценкой соответствующих структур в фазовом пространстве.Относительно недавно было установлено, что функциональный метод локализации позволяет анализировать поведение траекторий динамической системы. В частности, с помощью метода локализации можно проверять устойчивость положений равновесия.Здесь естественным образом возникает вопрос о связи функционального метода локализации с известным принципом инвариантности Ла-Салля, который можно рассматривать как дальнейшее развитие метода функций Ляпунова для установления устойчивости. Настоящая статья посвящена обсуждению этого вопроса

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Realization of the Iteration Procedure in Localization Problems of Autonomous Systems

In the last 15 years one way for a qualitative analysis of dynamical systems was formed i.e. the localization of invariant compact sets of a dynamical system. Here the localization means creating a system of such sets, which contain all invariant compact sets of a dynamic system [1], in the phase space.Invariant compact sets are closely connected with bounded trajectories of the system, the structure of which in the phase space play key role in many applications of dynamical system theory. The problems of invariant compact sets localization abut upon other important problems, for instance, the problems of estimation of attractor basins, control problems, etc.Back investigations of localization problems was oriented both to development of solving methods [28] and to investigation of particular dynamical systems encountered in applications (see, for example, [9 { 16]).One of quite efficient methods of localization problem solving is based on smooth functions defined in the phase space. It is so called functional method [1 { 3]. Effectiveness of the method is enhanced when we use several functions. Thus, using the next function gives the restriction of the already constructed localizing set. An iteration procedure for sequential narrowing of the localizing set [1 { 2] arises.The paper presents analysis of the iteration procedure, which naturally occur in the autonomous systems of special type where the right side of each differential equation is resolvable relative to the corresponding phase variable. Such systems are encountered in applications [17].</p

A Terminal Control Problem for the Second Order System with Restrictions

The paper considers a problem of the time-specified control terminal for the second order system with restrictions on the state variables.Most developed methods for solving problems of the terminal [1, 2, 3, 4, 5] do not allow us to take into account the restrictions on the system condition. To solve such problems are widely used methods based on the concept of inverse dynamics problems [6, 7, 8, 9, 10], with one step of which being to specify a kinematic object trajectory. Some methods use an iterative [11] process of finding a desired program trajectory.This work is based on the results presented in [12]. It is shown that the solution of the original problem is equivalent to finding the terminal phase of the trajectory that satisfies the restrictions imposed on the state variables, as well as the certain additional conditions. It is assumed that the restrictions imposed on the state variables can be represented as functions for which, in a certain class of functions, special approximations are built. A desired phase trajectory is built as a linear combination of obtained functions-approximations. Thus constructed phase trajectory is a solution to the original terminal problem. The presented formulas are true for both the upper and lower half-plane of the phase space. The paper proposes an optimization approach to a choice of the trajectory as well as the options to extend the set in which the phase trajectories are sought. It gives the numerical simulation results, a presented in [12] algorithm, and also the results of numerical solution to the optimization problem.This approach can be used to solve the terminal problems of vector-controlled mechanical systems with restrictions on the state variables.</p

Polynomials-Based Terminal Control of Affine Systems

One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of initial states such that the solution of the terminal control problem can be constructed by using a polynomial of degree 2n − 2.Note that solution of the terminal control problem in question can be used to solve the problem of stabilizing the zero equilibrium in a finite time.For the second-order systems we prove the necessary and sufficient conditions for existence of the polynomial of the second degree which determines the solution of the terminal problem. The solutions of the terminal control problem based on the polynomials of second and third degree are given. As an example, the terminal control problem is considered for the simple pendulum.We also discuss solution of the terminal problem for affine systems of the third order, based on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial state of the system and the origin are obtained.For systems of arbitrary order n we obtain the necessary and sufficient conditions for existence of a solution of the terminal problem using the polynomial of degree 2n − 2. We also give the solution of the problem by means of the polynomial of degree 2n − 1.Further research can be focused on extending the results obtained in this note to terminal control problems where the desired final state of the system is not necessarily the origin.One of the potential application areas for the obtained theoretical results is automatic control of technical plants like unmanned aerial vehicles and mobile robots.</p

Variations Method to Solve Terminal Problems for the Second Order Systems of Canonical Form with State Constraints

Terminal control problem with fixed finite time for the second order affine systems with state constraints is considered. A solution of such terminal problem is suggested for the systems with scalar control of regular canonical form.In this article it is shown that the initial terminal problem is equivalent to the problem of auxiliary function search. This function should satisfy some conditions. Such function design consists of two stages. The first stage includes search of function which corresponds the solution of the terminal control problem without state constraints. This function is designed as polynom of the fifth power which depends on time variable. Coefficients of the polynom are defined by boundary conditions. The second stage includes modification of designed function if corresponding to that function trajectory is not satisfied constraints. Modification process is realized by adding to the current function supplementary polynom. Influence of that polynom handles by variation of a parameter value. Modification process can include a few iterations. After process termination continuous control is found. This control is the solution of the initial terminal prUsing presented scheme the terminal control problem for system, which describes oscillations of the mathematical pendulum, is solved. This approach can be used for the solution of terminal control problems with state constraints for affine systems with multi-dimensional control.</p