Necessary and sufficient conditions of generalized exponential stability for retarded dynamic systems (Q2725358)

Here, necessary and sufficient conditions for the generalized exponential stability for a general type of retarded dynamic systems \(\dot x(t) = f(t,x_t)\) are established, where \(f:\mathbb{R}_+\times C_n \rightarrow \mathbb{R}^n\) is continuous and sufficiently smooth, \(C_n:=C([-\tau,0], \mathbb{R}^n)\) is supplied with the uniform norm \(\|\cdot\|_\tau\), \(\tau\geq 0\). Further, a generalized exponential stability result with respect to a time-varying decay degree for a special class of retarded dynamic systems is also presented based on the matrix measures as the application of the established condition. Here, the generalized exponential stability is understood according to the following two definitions: NEWLINENEWLINENEWLINEDefinition 1. A decay function is defined as \(d(t):[t_0,\infty) \rightarrow (0, \infty)\), \(t_0 \geq 0\), with \(\dot d(t) < 0\), where the dot denotes the right-hand derivative. If the solution \(x(t_0,\varphi)(t)\) to the system through any given \((t_0,\varphi) \in [0, \infty) \times C_n\) satisfies \(x(t_0,\varphi)(t) \leq d(t,\varphi)\) for all \(t\geq t_0\), then the decay gain of the function \(d(t,\varphi)\) is defined as \(\Gamma(t)=d(t,\varphi)/\|\varphi\|_\tau>0\), \(t\geq t_0\), with the initial decay gain \(\Gamma(t_0)\), and the decay degree of the function \(d(t,\varphi)\) is defined as \(\gamma(t)=-\dot d(t,\varphi)/d(t,\varphi)>0\), \(t\geq t_0\). NEWLINENEWLINENEWLINEDefinition 2. The retarded dynamic system is said to be globally generalized exponentially stable with respect to an initial decay gain \(\Gamma\) and decay degree \(\gamma(t)\) if the solution \(x(t_0,\varphi)(t)\) to the system through \((t_0,\varphi)\in [0,\infty)\times C_n\) satisfies NEWLINE\[NEWLINE \|x(t_0,\varphi)(t)\|\leq\Gamma\|\varphi\|_\tau \exp\Biggl\{-\int_{t_0}^t\gamma(t) dt\Biggr\}, NEWLINE\]NEWLINE where \(\Gamma\geq 1\) is a constant and \(\gamma:[0,\infty)\to(0,\infty)\) is a continuous positive function.