On the application of matrix-valued functionals in studying stability of systems with delay (Q2740277)

Here, the authors apply Lyapunov-Krasovskij matrix-valued functionals to obtain stability conditions for the trivial solution to systems with delay \(\dot x=f(t,x,x_t)\) where \(x_t=x_t(s)=x(t+s-\tau)\), \(s\in [-\tau,0]\), is considered as an element of the space \(C([-\tau,0] \mapsto \mathbb R^n)\) and \(f:\mathbb R_+\times \mathbb R^n\times C([-\tau,0] \mapsto \mathbb R^n) \mapsto \mathbb R^n\). In accordance with the decomposition of vector \(x\) into \(m\) subvectors \(x_i\in \mathbb R^{n_i}\), \(i=1,\ldots,m\), \(\sum n_i=n\), the matrix-valued functional has the form \(V(t,\varphi,\eta)=\eta^TU(t,\varphi)\eta \). Here, \(\eta \in \mathbb R^m\) is a vector with positive components \((\eta \in \mathbb R^m_{+})\), \(U(t,\varphi)=\{u_{ij}(t,\varphi)\}_{i,j}^m\), \(u_{ij}:\mathbb R_+\times C_H \mapsto \mathbb R\), and \(C_H\) stands for the \(H\)-neighborhood of zero in \(C([-\tau,0] \mapsto \mathbb R^n)\).NEWLINENEWLINENEWLINEUnder certain conditions imposed on \(V(t,\varphi,\eta)\) and its right Dini derivative \(D^+V\) along solutions to the system, the authors formulate Lyapunov-type theorems on stability, uniform stability, and uniform asymptotic stability of the trivial solution \(x\equiv 0\).