Method of Lyapunov functions in problems of stability of solutions of systems of differential equations with impulse action

Abstract

The resistivity and AC susceptibility of Hg3-aSbF6 cooled at different rates were measured between 1.4 and 4.2 K. There is a superconducting transition at 4.1 K in slowly cooled samples with no anisotropy in the superconducting transition temperature for current directions parallel and perpendicular to the c axis. Superconductivity of fast cooled samples occurs below 3.9 K and susceptibility changes over a range of temperature. The superconductivity which depends on cooling rate is attributed to mercury trapped in isolated regions, with the size of the regions depending on the rate of cooling.

Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

“…By virtue of condition (8), the segment OEt 0 ; t 0 C Â contains at most one point i : Therefore, kx.t; t 0 ; x 0 /k min.1; /kx 0 ke C 1 .t t 0 / for t 2 OEt 0 ; t 0 C Â:…”

“…Necessary conditions for the stability of solutions of systems with pulse action at fixed times were obtained in [6][7][8][9]. In the present paper, we establish necessary conditions for the stability of impulsive systems of a more general form, namely, systems with pulse action at nonfixed times.…”

On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations

“…In [3,5,10], under the assumption that conjectures C 1 -C 4 are true, conditions for the reversibility of the Gurgula -Perestyuk theorem on asymptotic stability were established.…”

On the stability in periodic and almost periodic difference systems

“…The systematic description of the theory of difference equations one can find in books [2,7,16]. Difference equations are a convenient model for discrete dynamic systems description and for mathematical simulation of systems with impulse effect [8,12,14,15,19]. One of directions arising from applications of difference equations is linked with qualitative investigation of their solutions (stability, boundedness, controllability, observability, oscillation, robustness.…”