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Zbl 1103.34054
A stability theory for second-order nonsmooth dynamical systems with application to friction problems.
(English)
[J] J. Math. Pures Appl. (9) 83, No. 1, 17-51 (2004). ISSN 0021-7824

An extension of the LaSalle invariance principle is developed for a large class of first-order evolution variational inequalities $$x(t) \in D( \partial \varphi )\quad\text{for }t \ge t_0,$$ $$\biggl\langle {dx \over dt}(t)+F(x(t)),v-x(t) \biggr\rangle + \varphi (v) - \varphi (x(t)) \ge 0\quad\text{for }v \in \Bbb R^n \text{ and a.a. }t \ge t_0 , x(t_0) = x_0,$$ and applied to study stability and asymptotic properties of the important classes of second-order dynamic systems $$H_2 {dq \over dt} (t) \in D( \partial \Phi )\quad\text{for }t\ge t_0,$$ $$M { d^2 q \over dt^2} (t) + C {dq \over dt }+ K q(t) \in - H_1 \partial \Phi \biggl(H_2 {dq \over dt} (t) \biggr),\quad\text{ for a.a. }t \ge t_0.$$ The term $H_1 \partial \Phi$ is introduced in order to model the unilaterality of the contact induced by friction forces together with some illustrative small-sized examples in mechanics.
[Tran Nhu Pham (Hanoi)]
MSC 2000:
*34G25 Evolution inclusions
47J20 Inequalities involving nonlinear operators
49J40 Variational methods including variational inequalities
35K85 Unilateral problems; variational inequalities (parabolic type)
74M10 Friction
34D20 Lyapunov stability of ODE
70F40 Problems with friction

Keywords: Variational inequalities; unilateral dynamic systems; nonsmooth mechanics; Lyapunov stability; differential inclusion, LaSalle invariance principle; convex analysis; friction problems in mechanics

Cited in: Zbl 1239.34070

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