On instability of conservative systems with gyroscopic forces

Theorems on equilibrium instability of conservative systems with gyroscopic forces are proved. The theorems obtained are nonlinear analogs of the Kelvin theorem. The equilibrium instability of the Chaplygin nonholonomic systems is considered. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1422–1428, October, 1997.     

On instability under small periodic perturbation of two-dimensional hyperbolic systems

Under consideration is a mixed problem for a strictly hyperbolic linear first-order system in the half-strip Π = {(x, t): 0 < x < 1, t > 0} generating a group of unitary operators. In the case of a periodic perturbation a method is proposed for finding the frequencies for which the perturbed system develops parametric resonance. The method is illustrated with a system of two equations.     

On the gyroscopic stabilization of conservative systems

We consider conservative systems with gyroscopic forces and prove theorems on stability and instability of equilibrium states for such systems. These theorems can be regarded as a generalization of the Kelvin theorem to nonlinear systems.     

On stability boundaries of conservative systems

Stability boundaries of linear conservative systems smoothly dependent on several parameters are studied. Generic singularities appearing on the stability boundaries are classified. Explicit formulae for the approximations to the stability domain at regular and singular points of the boundary are derived. These formulae use information on the system only at the point under consideration (eigenvectors and derivatives of the stiffness matrix with respect to parameters). As an example a buckling problem of a column loaded by an axial force is considered and discussed in detail.     

On the stability of linear conservative gyroscopic systems

A simple criterion concerning the stability of linear gyroscopic conservative systems is developed. First, a sufficient condition for stability is established. The proof is based on a transformation converting the original autonomous system into a nonautonomous system, and applying Liapunov's direct method to the latter. Then a theorem is given which provides a necessary and sufficient condition for a restricted class of systems. Illustrative examples are provided.     

On the observability of time-discrete conservative linear systems

We consider various time discretization schemes of abstract conservative evolution equations of the form , where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair (A,B) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443-471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.     

On instability of the equilibrium state of nonholonomic systems

We establish a criterion of instability for the equilibrium state of nonholonomic systems, in which gyroscopic forces may dominate over potential forces. We show that, similarly to the case of holonomic systems, the evident domination of gyroscopic forces over potential ones is not sufficient to ensure the equilibrium stability of nonholonomic systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 389–397, March, 1999.     

On the instability of differential systems with varying delay

The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r₁(t))+f(t,x(t),x(t−r₂(t))) are examined; the nonconstant delays r₁, r₂ are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained.     

Toric degenerations of Gelfand-Cetlin systems and potential functions

We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system.     

On the instability of non-semi-Fredholm operators under compact perturbations

The stability of several natural sets of the non-semi-Fredholm operators in a separable Hilbert space under compact perturbations studied by R. Bouldin. (The instability of non-semi-Fredholm operators under compact perturbations, J. Math. Anal. Appl.87 (1982), 632–638.) The aim of the present article is to study this problem in arbitrary Banach spaces. We also derive a curious characterization of separable Banach spaces.     

On the geometry of quadrics and their degenerations

members of G.N.S.A.G.A. of C.N.R.     

On the asymptotic stability of equilibria of nonlinear mechanical systems with delay

We consider some classes of nonlinear mechanical systems with retarded argument. It is assumed that, in the absence of delay, the systems in question have asymptotically stable equilibria. We analyze how the delay affects the stability of these equilibria. The Lyapunov function method and Razumikhin’s approach are used to derive conditions under which asymptotic stability is preserved for arbitrary delay values. We suggest a method for stabilizing strongly nonlinear conservative systems by constructing a delay feedback control depending only on the generalized coordinates.     

On periodic asymptotic equilibria of systems of nonlinear finite-difference equations

We obtain new sufficient conditions for the existence of periodic asymptotic equilibria of systems of nonlinear finite-difference equations with continuous argument.