Asymptotic stability for damped strongly nonautonomous systems with applications to holonomic mechanical systems and nonlinear networks (Q1386392)

The authors deal with the asymptotic stability of the solutions to ordinary differential systems with time dependent restoring potential of the type: \[ (\nabla_{p}G(u,u')) - \nabla_{u}G(u,u') + q^{m}(t)f(u)=Q(t,u,u'),\tag{1} \] with \(J=[t,\infty), u : J \rightarrow {\mathbb{R}^{N}}\), and roughly, \(G=G(u,p)\) is strictly convex and homogeneous of degree \(m >1\) in \(p\) for all \(u,q\) is a \(C^{1}\) positive function, \(f\) is a continuous restoring term arising from a potential \(F\), that is \(f=\nabla_{u}F\) and \((f(u),u) > 0\) for \(u \neq 0\), and \(Q\) is a continuous damping term defined on \(J \times {\mathbb{R}^{N}} \times {\mathbb{R}^{N}}\), i.e. \((Q(t,u,p),p) \leq 0\), where \((\cdot,\cdot)\) denotes the scalar product in \({\mathbb{R}^{N}}\). The asymptotic stability of the solutions to (1) has been widely studied in the literature in particular cases: \[ u''(t) + h(t)|u|^{\beta} u' + q^{2}(t)f(u)=0, t \in J,\tag{2} \] for \(\beta = 0\), holonomic bilaterally constrained mechanical systems and nonlinear networks of circuits, etc. Recently, \textit{P. Pucci} and \textit{J. Serrin} [Arch. Ration. Mech. Anal. 132, No. 3, 207-232 (1995; Zbl 0861.34034)] developed a fairly general asymptotic stability theory for ordinary differential systems of type (1), containing known results for several problems of this type as particular cases, assuming that the nonlinear damping magnitude \(|Q|\) be controlled from below by a function of the form \[ \delta(t) \Phi(p), \Phi (p) > 0 \text{ for } p \neq 0\tag{3} \] and the geometrical condition on \(Q\): for all \(t \in J, |u|\leq U\) and \(p \in {\mathbb{R}^{N}}\) \[ |Q(t,u,p)|\cdot |p|\leq \gamma|Q(t,u,p),p)|, \text{ some } \gamma \geq 1.\tag{4} \] The authors extend Pucci and Serrin's results to the case in which: i) \(Q\) is controlled by functions of the type \(\delta(t)\Phi(u,v)\), with \(\Phi (u,v) > 0\) for \(u \neq 0\) and \( v \neq 0\), instead of (3), and ii) \(Q\) does not satisfy the geometric condition (4), but a weaker condition introduced also by Pucci and Serrin. The first generalization allows to extend their results to holonomic mechanical systems, for which \(Q(t,u,p)=L(t,u)p\) with matrices \(L(t,u)\) positive definite, but possibly not uniformly positive definite in \( J \times {\mathbb{R}^{N}}\) and equations as (2) with \(\beta \neq 0\). \[ u''(t) + h(t)|u|^{\beta} u' + q^{2}(t)f(u)=0, t \in J. \] The second generalization allows to extend the stability results given by \textit{R. J. Duffin} [J. Math. Anal. Appl. 23, No.~2, 428-439 (1968; Zbl 0167.37701)] concerning nonlinear networks of circuits to very general nonautonomous systems. So \(Q\) may depend in an essential way on the variable \(u\) and can be not tame, a geometrical condition required in previous works.