Convergent solutions of ordinary and functional-differential pendulum- like equations (Q1203058)

The system (1) \(dz/dt=Ax+b\varphi(\sigma)\), \(d\sigma/dt=c^* z+\rho\varphi(\sigma)\) is considered. Here \(z\in\mathbb{R}^ n\), \(\sigma\in\mathbb{R}^ 1\), \(A\), \(b\), \(c\) are constant matrices and vectors, \(A\) is a stable matrix, \((A,b)\) and \((A^*,c)\) are controllable, \(\rho\) is a constant, \(\rho\neq c^* A^{-1} b\), \(\varphi\) is a continuously differentiable \(\Delta\)-periodic function, and \(\mu_ 1\leq\varphi'(\sigma)\leq\mu_ 2\) for all \(\sigma\in\mathbb{R}'\). Let \(K(\rho)=-\rho+c^*(A-\rho I_ n)^{-1} b\) be the transform function from \(\varphi\) to \((-\sigma)\). A typical result: suppose that there exist numbers \(\delta>0\), \(\varepsilon>0\), \(\tau\geq 0\) such that the inequality \(\text{Re}\{K(i\omega)-\tau [K(i\omega)-\mu_ 1^{- 1}i\omega]^* [K(i\omega)-\mu_ 2 i\omega]\} -\varepsilon| K(i\omega)|^ 2\geq\delta\) holds for all \(\omega\in\mathbb{R}'\) and at least one of the conditions \[ \begin{aligned} 2\sqrt {\tau\delta} &> \int_ 0^ \Delta \varphi(\sigma)d\sigma\cdot \left[ \int_ 0^ \Delta \sqrt {(1- \mu_ 1^{-1} \varphi'(\sigma))(1-\mu_ 2^{-1} \varphi'(\sigma))} | \varphi(\sigma)| d\sigma\right]^{-1},\tag{i}\\ 2\sqrt{\varepsilon\delta} &> \int_ 0^ \Delta \varphi(\sigma)d\sigma\cdot \left[ \int_ 0^ \Delta | \varphi(\sigma)| d\sigma\right]^{-1}\tag{ii}\end{aligned} \] is fulfilled. Then all solutions \(x(t)=[z(t),\sigma(t)]\) of (1) are convergent: \(x(t)\to x_ \infty\) as \(t\to+\infty\) where \(x_ \infty\) is an equilibrium point.