Exponential stability of nonlinear time-varying differential equations and applications (Q2718931)

The following nonlinear system described by the time-varying differential equations NEWLINE\[NEWLINE\dot x(t)=f(t,x(t)),\;t\geq 0;\;x(t_0)=x_0,\;t_0\geq 0,\tag{1}NEWLINE\]NEWLINE with \(x(t)\in\mathbb{R}^n\), \(f(t,x): \mathbb{R}^+\times \mathbb{R}^n\to \mathbb{R}^n\), \(f(t,0)=0\) \(\forall t \in \mathbb{R}^+\), is studied. Associated with (1) a nonlinear time-varying control system NEWLINE\[NEWLINE\dot x(t)=f\bigl(t,x(t),\;u(t)\bigr),t\geq 0,\tag{2}NEWLINE\]NEWLINE with \(x\in\mathbb{R}^n\), \(u\in\mathbb{R}^m\), \(f(t,x,u):\mathbb{R}^+ \times\mathbb{R}^n \times\mathbb{R}^m \to\mathbb{R}^n\), is considered.NEWLINENEWLINENEWLINEThe main results of this paper are:NEWLINENEWLINENEWLINE(i) New sufficient conditions for the exponential stability of the zero solution to (1) by the Lyapunov-like functions are given. (A function \(V(t,x): W\to\mathbb{R}\) is called a Lyapunov-like function for (1) (a generalized Lyapunov-like function for (1)) if \(V(t,x)\) is continuously differentiable in \(t\in\mathbb{R}^+\) and in \(x\in D\) (if \(V(t,x)\) is continuous in \(t\in\mathbb{R}^+\) and Lipschitzian in \(x\in D\), uniformly in \(t)\), and there exist positive numbers \(\lambda_1,\lambda_2,\lambda_3\), \(k,p, q,r,\delta\) such that \(\lambda_1\|x\|^p \leq V(t,x) \leq\lambda_2 \|x\|^q\), \(\forall(t,x)\in W\); \(D_fV(t,x) \leq-\lambda_3 \|x\|^n +K e^{-\delta t}\), \(\forall t\geq 0\), \(x\in D\setminus \{0\}\) (and there exist positive functions \(\lambda_1(t)\), \(\lambda_2(t)\), \(\lambda_3(t)\), where \(\lambda_1 (t)\) is nondecreasing, and there exist positive numbers \(K,p,q,r, \delta\) such that NEWLINE\[NEWLINE\lambda_1 (t)\|x\|^p\leq V(t,x)\leq \lambda_2(t) \|x\|^q,\;\forall(t,x)\in W,NEWLINE\]NEWLINE NEWLINE\[NEWLINED_f^+V(t,x) \leq-\lambda_3 (t)\|x\|^r +Ke^{-\delta t},\forall t\geq 0,\;x\in D\setminus \{0\}).NEWLINE\]NEWLINE (ii) New sufficient conditions for (1) by the generalized Lyapunov-like function are given.NEWLINENEWLINENEWLINEThese results are applied to some stabilization problems of nonlinear time-varying control system (2).