Uniform global asymptotic stability of differential inclusions
The authors consider the stability of differential inclusions of the type \[ x'(t)\in F(x(t)), \tag{1} \] where the state \(x(\cdot)\) evolves in \(\mathbb{R}^n\), and the set-valued function \(F\) is locally Lipschitz and takes values which are nonempty compact subsets of \(\mathbb{R}^n\). They denote by \(S\) the set of all maximal solutions of (1) and their domains by \([0,T_{x(\cdot)}^{\max})\). Then for each \(x(\cdot)\in S\), the output signals are denoted as \(y(t)=h(x(t))\), \(w(t)=k(x(t))\), where \(h\) and \(k\) are given continuous functions. The differential inclusion (1) is supposed to satisfy the unbounded observability property (through w), i.e., for each \(x(\cdot)\in S\) such that \(T_{x(\cdot)}^{\max}<\infty\), we have \[ \limsup_{t\to T_{x(\cdot)}^{max}}| w(t)| =\infty. \] The main result for such differential inclusion is that (1) satisfies the global weak error-attractivity modulo measurements property if and only if it satisfies the uniform global weak error-attractivity modulo measurements property. The definitions of these notions are as follows. Global weak error-attractivity modulo measurements: There exists some continuous, positive definite and strictly increasing function \(\gamma\) such that the function \( z(t)=\max\{| y(t)| -\gamma(| w(t)| ), 0\}\) satisfies for all \(x(\cdot)\in S\) \[ \inf_{0\leq t\leq T_{x(\cdot)}^{\max}}| z(t)| =0. \] Uniform global weak error-attractivity modulo measurements: For \(z\) defined above, for any \(\varepsilon>0\) and any \(\kappa>0\) there exists \(T_{\varepsilon, \kappa}\) such that for any \(\xi \in \mathbb R^n\) with \(| \xi| \leq \kappa\) and any \(x(\cdot)\in S\) starting at \(\xi\), if \(T_{\varepsilon,\kappa} <T^{\max}_{x(\cdot)}\), then there exists some \(\tau<T_{\varepsilon,\kappa}\) such that \(| z(\tau)| \leq \varepsilon.\)
scientific article; zbMATH DE number 2091634