Conditions for practical \(\mu\)-stability of quasilinear systems with lags (Q2714580)

The authors deal with the quasilinear system of differential equations with delay NEWLINE\[NEWLINE \frac{dx}{dt} = A(t)x(t) +\mu R(t,x,x_t),\tag{1} NEWLINE\]NEWLINE where \(A(t)\) is an continuous \(n\times n\)-matrix on \([0,\infty)\); \(R: \mathbb{R}_+\times \mathbb{R}^n\times C_H\to \mathbb{R}^n\), \(C_H\) is a set \(\varphi\in C\) for which \(\|\varphi\|\leq H\); \(C=C([-\tau,0],\mathbb{R}^n)\), \(\|\varphi\|_0=\max_{-\tau\leq s\leq 0} |\varphi(s)|\). For the given domains \(S_\alpha\) and \(S_\beta\); \(S_\alpha\subset S_\beta\) and \(\partial S_\alpha\cap \partial S_\beta=\emptyset\) the following definition is formulated: system (1) is practically \(\mu\)-stable, if for any \(\varphi(\theta)\in S_\alpha\) there exists \(\mu_0>0\) such that \(x(t_0,\varphi,\mu)(t)\in\text{int} S_\beta(t)\) for \(0\leq\mu<\mu_0\) and all \(t\geq t_0\).NEWLINENEWLINENEWLINENew conditions for the practical \(\mu\)-stability of various types are established in terms of the comparison system constructed to system~(1).