On instability of solutions to an uncertain system with respect to a given moving set (Q2740281)

The author considers the system of the formNEWLINE\[NEWLINE\dot x=f(t,x,\alpha),\quad f:[t_0,\infty)\times \mathbb R^n\times \mathbb R^d \mapsto \mathbb R^n.\tag{1}NEWLINE\]NEWLINE Here, the parameter \(\alpha \) is treated as a measure of the system uncertainty. The author studies the problem about the instability of the set \(A(\alpha):=\{x\in \mathbb R^n:\|x\|=r(\alpha)\}\) where \(r(\alpha)\) is a positive function such that \(r(\alpha)\to r_0>0\) as \(\|\alpha \|\to 0\) and \(r(\alpha)\to \infty \) as \(\|\alpha \|\to \infty \).NEWLINENEWLINENEWLINEInstability conditions are obtained in the framework of the direct Lyapunov method by using the Lyapunov function of the form \(V(t,x,y):=y^TU(t,x)y\) [see also the author's paper in Ukr. Math. J. 51, No. 4, 508-517 (1999); translation from Ukr. Mat. Zh. 51, No. 4, 458-465 (1999; Zbl 0940.34045)]. Here, \(U(t,x)=\{u_{ij}(t,x)\}_{i,j=1}^s\), and \(y\in \mathbb R^s\) is an appropriate vector. The author requires the right upper Dini derivative \(D^+V|_{(1)}\) to be positively (negatively) definite outside (inside) the sphere \(A(\alpha)\), and \(D^+V|_{(1)}=0\) for \(\|x\|=r(\alpha)\). It should be noted that the matrix \(U(t,x)\) does not appears explicitly nor in formulation of the main result neither in its proof.