Controllability and stabilization of programmed motions of reversible mechanical and electromechanical systems

Problems of controllability and methods of stabilizing programmed motions of a large class of mechanical and electromechanical systems which are reversible with respect to the control are considered. Criteria of the controllability and stabilizability of reversible systems are obtained. Programmed motions and algorithms of programmed control are designed in analytical form and algorithms of programmed motions for non-linear reversible systems are synthesized.     

Reverse motions of mechanical systems

The possibility of the occurrence of sections of reverse motions in natural mechanical systems, when, in the second half of a time interval, the motion in the first half of the interval is repeated in the reverse order and the opposite velocity with a specified accuracy, is investigated. It is shown that such motions are characteristic of natural mechanical systems in the neighbourhood of a non-degenerate equilibrium position if the natural frequencies are independent. Systems with gyroscopic and dissipative forces are also considered. It is shown that, in these systems, sections of reverse motion can be observed in a special system of coordinates. Examples are presented.     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p ₀ of a Z ₂(ℝ)-reversible vector field in ℝ²ⁿ with reversing symmetry R satisfying R ² = I and dimFix(R) = n. We deal with one-parameter families of such systems X _λ such that X ₀ presents at p ₀ a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X ₀ (at p ₀) has as eigenvalues: λ₁ = 0 and λ _j = ±iα _j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.     

Dissipative periodic systems and symmetric interpolation in Schur classes

Matrix-valued functions analytic and contractive in the open unit disk (Schur functions) play an important role in system theory. They represent transfer functions of causal time-invariant dissipative systems. In this paper we show how a Schur function can still be associated to a k-periodic system. This function satisfies a certain symmetry condition (conditions (1.7) for k=2 and (4.4) for general k). We study a general bitangential interpolation problem for the Schur functions satisfying this symmetry condition.     

Stabilization of the motions of non-autonomous mechanical systems

The problem of stabilizing the motions of mechanical systems that can be described by non-autonomous systems of ordinary differential equations is considered. The sufficient conditions for stabilizing of the motions of mechanical systems with assigned forces due to forces of another structure are obtained by constructing a vector Lyapunov function and a reference system. Examples of the solution of the problems of stabilizing the rotational motion of an axisymmetric satellite in an elliptic orbit, a non-tumbling gyro horizon, etc. are considered ©2009     

Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term

In this paper, the boundedness of all solutions of the oscillator

x^″+f(x,x^′)+ω²x+?(x)=p(t)$x^{″}+f(x,x^{\prime })+{\omega }^{2}x+?(x)=p(t)$

Stationary and nonstationary stability of periodic solutions of reversible systems

M. V. Lomonosov Moscow State Univeristy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 2, pp. 40–48, April–June, 1989.     

Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

This paper analyses the existence of invariant manifolds ofperiodic orbits for a specific piecewise linear three-dimensionalsystem with two zones, whose linear parts share a pair of imaginaryeigenvalues. This degenerate situation is obtained from thelack of controllability. The analysis proceeds by its reductionto a periodic one-dimensional equation for which some resultsof the Ambrosetti–Prodi type are given.     

Stability of the periodic motions of autonomous systems with lag

In the article we give a generalization of the theorem of A. A. Andronov and A. A. Vitt on the stability of the periodic motions of autonomous systems described by ordinary differential equations to autonomous time-lag systems. The investigation is based on the procedures developed by N. N. Krasovskii which consider the differential equations with lag in a functional space of continuous functions. There is a bibliography of eight items.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp.561–568, November, 1967.     

Invariant sets and Knaster-Tarski principle

Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.     

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.     

Invariant solutions of rank two of the equations of the rotationally symmetric motions of an inhomogeneous liquid

An optimal system of first-order algebras of the system of equations for the rotationally-symmetric unsteady motion of an inhomogeneous liquid is constructed. New exact solutions of certain factor systems are found which describe motions with free boundaries or internal non-linear waves.     

Special symmetric periodic solutions of differential systems with distributed delay

We discuss the existence of periodic solutions to a system of differential equations with distributed delay which shows a certain type of symmetry. For this, such solutions are related to the solutions of a system of second-order ordinary differential equations.