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Том 52, № 11 (2016)
- Год: 2016
- Статей: 12
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9285
Ordinary Differential Equations
Sign reversal of the characteristic exponents of solutions of a differential system with initial data on finitely many points and lines
Аннотация
We realize a version of the Perron sign reversal effect for the characteristic exponents of a two-dimensional differential system; the exponents are negative for the linear approximation system and positive for the nontrivial solutions of the full nonlinear system with a higher-order perturbation in a neighborhood of the origin and with initial data on an arbitrary finite set of points and lines on the plane R2.
1389-1402
Global asymptotic stability analysis by the localization method of invariant compact sets
Аннотация
We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of the localization method for invariant compact sets of autonomous systems. These results are related to finding sets that contain all invariant compact sets of the system (localizing sets) and to the behavior of trajectories of the system with respect to localizing sets. We consider an example of a system whose equilibrium belongs to the critical case.
1403-1410
Control Theory
Properties of extremals in optimal control problems with state constraints
Аннотация
An optimal control problem with state constraints is considered. Some properties of extremals to the Pontryagin maximum principle are studied. It is shown that, from the conditions of the maximum principle, it follows that the extended Hamiltonian is a Lipschitz function along the extremal and its total time derivative coincides with its partial derivative with respect to time.
1411-1422
Symmetries, coverings, and decomposition of systems and trajectory generation
Аннотация
We derive relations between the notions of symmetry, covering, and decomposition of systems and trajectory generation. We show that any decomposition of a system determines a finite-dimensional covering of that system and is determined by it. We present conditions on vector fields under which any covering is obtained by factorization along the Lie algebra of such fields. On the basis of these relations, we study whether a point-to-point steering problem can be transformed into a set of boundary value problems of lower dimension.
1423-1435
Construction of almost integral manifolds of affine control systems
Аннотация
We consider the problem of constructing manifolds of a special form, which are said to be almost integral, lie in the state space of an affine control system, and have the following property: the restriction of the control system to such a manifold is an almost trivial control system defining the part of trajectories of the original system lying on that manifold.
1436-1441
Guidance problem for a distributed system with incomplete information on the state coordinates and an unknown initial state
Аннотация
We study the problem of guaranteed positional guidance of a linear partially observable control system with distributed parameters to a convex target set at a given time. The problem is considered under incomplete information. More precisely, we assume that the system is subjected to an unknown disturbance; in addition, the initial state is assumed to be unknown as well. Further, the sets of admissible disturbances and the set of admissible initial states, which is assumed to be finite, are known. An algorithm for solving the problem is suggested.
1442-1452
Criteria for modal controllability of linear systems of neutral type
Аннотация
For linear autonomous differential-difference systems of neutral type with commensurable delays, we suggest solvability criteria for the modal controllability problem with the use of two classes of state feedback controllers, namely, controllers of constant and variable structure. The proofs are constructive and permit one to obtain the corresponding controllers with the use of standard operations on polynomials and polynomial matrices.
1453-1468
Notions of equilibrium for differential games on intersecting game sets
Аннотация
We suggest new notions of conflict equilibrium and demonstrate a technique of their use for finding a solution in arbitrary game problems on a game set common for all players and especially in problems with side interests of players in the static and dynamic settings.
1469-1480
On the properties of zero dynamics of linear systems
Аннотация
We consider a linear time-invariant multivariable square control system. For this system, we are interested in the description of zero dynamics, that is, the dynamics of the system for the case of identically zero output. We study the case in which the relative degree is undefined for the system.
1481-1493
Delay stability margins of linear plants
Аннотация
For linear plants with indefinite delay, we introduce the notion of delay stability margin and study the problem of finding the maximum delay stability margin for linear dynamic plants with respect to a given set of stabilizing feedbacks.
1494-1503
Numerical Methods
Regularized extragradient method in multicriteria control problems with inaccurate data
Аннотация
The class of Hilbert space multicriteria optimization problems considered in the paper includes control problems for various dynamical systems with lumped as well as distributed parameters. An equilibrium point is sought under the assumption that the criteria and their derivatives are known approximately. We use a regularized extragradient method and prove its convergence. As a sample application of the general theory, we consider a control problem for a parabolic equation with two criteria.
1504-1516
Short Communications
On the spectral theory of the Bessel operator on a finite interval and the half-line
Аннотация
We study the minimal and maximal operators generated by the Bessel differential expression on a finite interval and the half-line. We describe the domains of the Friedrichs and Krein extensions of the minimal operator and all nonnegative self-adjoint extensions of the minimal operator.
1517-1522