The asymptotic motions of systems with dissipation

On the stability of equilibria of nonholonomic systems with nonlinear constraints

“…In accordance with the major results from [2][3][4][5][6][7], referring to the generalization of Lyapunov's first method to the strongly nonlinear systems of differential equations, further considerations are based on the following statement: if differential equations (2) and (5) allow the existence of the solution q = q(t) with the property q (t) → 0 when t → −∞, then the equilibrium position q = 0,q = 0 is unstable.…”

“…If that series exists and is divergent, then, as shown in [9], there exists the solutionq =q(t) of (2) and (5) for which the series (13) represents the asymptotic presentation. It is inferred that the existence of the series (13) entails the instability of the position of equilibrium q = 0,q = 0 of the system, whose motion is described by the differential equations (2) and (5).…”

“…In the case with m > 2 (see (7)), the solution q = q (t) with the property q (t) → 0 when t → −∞ is sought in the form of the infinite series, possibly also divergent [5][6] ,…”

“…To find the conditions for the existence of the series given above, we take [5][6] a 1 = λe, where λ > 0, e = (e 1 , · · · , e n ). The series (13), for the case with μ = 2 m−2 , incorporated in (11) and (12), yields the following relations (only terms relevant to further analysis are shown),…”

“…In the three cases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints ) is generalized to the case of nonlinear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov (1994) to nonholonomic systems with nonlinear constraints. …”