Nonuniform asymptotic stability of nonlinear dynamic systems: A study based on the symmetry of their properties (Q1284301)

This article is a sequel to the article of the author in the same journal [\((*)\) \textit{V. P. Zhukov}, Autom. Remote Control 57, No. 6, Pt. 1, 801-807 (1996); translation from Avtom. Telemekh. 1996, No. 6, 40-48 (1996; Zbl 0932.93076)] on the subject of asymptotic stability of equilibrium states of dynamic systems described by non-autonomous ordinary differential equations. The system investigated is \[ d{\mathbf x}/dt= f({\mathbf x},t),\quad {\mathbf x}\in\mathbb{R}^n,\quad f(\mathbf{0},t)= 0.\tag{1} \] The function \(f({\mathbf x},t)\) is continuous in a region \(G_p= G\times\mathbb{R}^+\) where \(G\subset \mathbb{R}^n\) is a neighborhood of the isolated equilibrium point \({\mathbf x}= \mathbf{0}\) and \(\mathbb{R}^+= [0,\infty)\). The author studies stability of the equilibrium at the origin without making any uniformity assumptions, regarding how the assignment of \(t_0\), \({\mathbf x}_0\) affects the solution \({\mathbf x}(t, t_0,{\mathbf x}_0)\) as was done by \textit{N. N. Krasovskij} in his 1959 monograph [Some problems in the theory of stability of motion (in Russian) (Fizmatgiz, Moskva) (1959; Zbl 0085.07202)]. Uniform stability is referred to as R-stability, and non-uniform stability as N-stability. Uniform stability implies that \(\lim_{t\to\infty}{\mathbf x}(t,t_0,{\mathbf x}_0)= 0\) holds uniformly in both \(t_0\) and \({\mathbf x}_0\). The C-stability implies uniformity with respect to \(t_0\) of the stability of the point \({\mathbf x}= \mathbf{0}\) in the following sense: the system is C-stable if the point \({\mathbf x}= \mathbf{0}\) is uniformly stable with respect to the choice of \(t_0\), implying that given an \(\varepsilon\)-neighborhood of the origin there exists a number \(\delta(\varepsilon)\) independent of \(t_0\) such that for any \((t_0,{\mathbf x}_0)\in \delta\)-neighborhood of \(\mathbf{0}\times \mathbb{R}^+\) it is true that \({\mathbf x}(t, t_0,{\mathbf x}_0)\) stays in the \(\varepsilon\)-neighborhood of \(\mathbf{0}\) for all values of \(t\). Also the origin is called R-stable if it is asymptotically stable and \(\lim_{t\to\infty}{\mathbf x}(t, t_0,{\mathbf x}_0)= 0\) holds uniformly in both \(t_0\) and \({\mathbf x}_0\) and there exists a neighborhood \(A\) of \({\mathbf x}=\mathbf{0}\) such that given an arbitrary neighborhood \(\eta\) of \({\mathbf x}= \mathbf{0}\), it is possible to find a number \(T(\eta)\) such that for any choice of \((t_0,{\mathbf x}_0)\in A\times \mathbb{R}^+\) the trajectory \({\mathbf x}(t, t_0,{\mathbf x}_0)\) stays in the \(\eta\)-neighborhood of \(\mathbf{0}\) for all values of \(t\in [t_0+ T(\eta),\infty)\). The origin \({\mathbf x}= \mathbf{0}\) is called AC-stable if for any \(\varepsilon\)-neighborhood of the origin there exists a neighborhood \(\delta(\varepsilon)\) independent of \(t_0\) such that \({\mathbf x}(t,t_0,{\mathbf x}_0)\in \varepsilon\)-neighborhood of the origin for all \(t\in [t_0,\infty)\), and \(\lim_{t\to\infty}{\mathbf x}(t,t_0,{\mathbf x}_0)= \mathbf{0}\) holds for any \(({\mathbf x}_0, t_0)\in \delta(\varepsilon)\times\mathbb{R}^+\). An AC-system that is also an N-system is called an NC-system. An RC-system is defined similarly. In his article \((*)\) the author introduced a novel approach based on a nonlinear change of the time scale \(t(\tau)\) which produces a symmetry with respect to some dynamic property of the system, thus generating a conservation law. This transformation maps \(\mathbb{R}^+\) onto \(\mathbb{R}^+\) and is invertible, and \(\lim_{\tau\to\infty} t(\tau)= \infty\), \(\lim_{t\to\infty}\tau(t)= \infty\), and if \(\tau_2> \tau_1\) then \(t(\tau_2)> t(\tau_1)\). Such functions \(\{t(\tau), \tau(t)\}\) are called admissible. The state equation in the \(\tau\)-scale becomes \[ d\xi/d\tau= f(\xi,t(\tau)) dt/d\tau,\quad \xi(\tau)= {\mathbf x}(t(\tau)).\tag{2} \] The author proves that properties of the points of equilibrium related either to trajectories or to a phase space portrait, stated for \(\xi=\mathbf{0}\), are unchanged when stated for \({\mathbf x}= \mathbf{0}\). Such properties could be stability, or instability, or some specific form of stability, such as AC-stability, or the number of attractive regions, etc. The main theorem states that the equilibrium point \({\mathbf x}= \mathbf{0}\) for system (1) is AC-stable if and only if there exists a function \(t(\tau)\) transforming system (1) into system (2) such that for the equilibrium point \(\xi= \mathbf{0}\) there exists a function satisfying Lyapunov's theorem on asymptotic stability. The proof is lengthy and rather difficult. The author constructs the function \(T(\eta, t)\) in steps taking several pages, arguing about behavior of the trajectory along small segments embedded in compact spheres of radius \(r_\eta\). This is a subtle argument, where at first one is convinced that at least two continuous derivatives shall be required to come up with a proof, and that perhaps only existence of the crucial function \(T(\eta)\) can be proved. The author completes this difficult construction using an auxiliary step function. The last sentence of his proof states that the construction of the function \(t(\tau)\) reveals that it needs to be only once continuously differentiable. One expects a high level of competence from members of the Krasovskij school, and this article confirms it.