Hyperbolic control sets and chain control sets
In the paper control-affine systems of the form \[ \dot x(t)= X_0(x(t))+ \sum^m_{i=1} u_i X_i(x(t)),\quad t\in \mathbb{R},\tag{1} \] \[ u\in{\mathcal U}= \{u:\mathbb{R}\to \mathbb{R}^m,\;(u_i(t))\in U\text{ for almost all }t\in\mathbb{R}\}, \] are considered. Here \(X_0,\dots, X_m\) are vector fields on a \(d\)-dimensional Riemannian manifold \(N\) and \(U\in \mathbb{R}^m\). It is assumed that \(\forall(u,x)\in{\mathcal U}\times N\) there is a unique solution \(\varphi(t,x,u)\), \(t\in\mathbb{R}\), of (1) with the initial condition \(\varphi(0,x,u)= x\) and control \(u\). \(D\subset N\) is a control set if 1) \(\forall x\in D\), \(D\subset\text{cl }O(x)\), 2) \(\forall x\in D\), \(\exists u\in{\mathcal U}: \varphi(t,x,u)\in D\), \(\forall t\geq 0\), 3) \(D\) is maximal with these properties. For fixed \(x,y\in N\), \(\varepsilon, T>0\), a controlled \((\varepsilon,T)\)-chain from \(x\) to \(y\) is given by \(x= x_0,x_1,\dots, x_n= y\in N\); \(u_0,\dots, u_{n-1}\in{\mathcal U}\), \(t_0,\dots, t_{n-1}\geq T\) and \(d(\varphi(t_j, x_j, u_j), x_{j+1})\leq \varepsilon\), \(j= 0,\dots, n-1\). \(E\subset N\) is called a chain control set of (1) if 1) \(\forall x\in E\), \(\exists u\in{\mathcal U}: \varphi(t,x,u)\in E\), \(\forall t\in \mathbb{R}\), 2) \(\forall x,y\in E\), \(\varepsilon,T> 0\) there is a controlled \((\varepsilon, T)\)-chain from \(x\) to \(y\), 3) \(E\) is maximal with these properties. For a chain control set of \(E\), its lift to \({\mathcal U}\times N\) is defined by \({\mathcal E}= \{(u,x)\in{\mathcal U}\times N\), \(\varphi(t,x,u)\in E\), \(\forall t\in \mathbb{R}\}\). Some shadowing property (condition (S)) is defined. The main results of the paper: Assume that system (1) is locally accessible and consider a chain control set \(E\) with nonvoid interior. If condition (S) holds for the lift \({\mathcal E}\) of \(E\), then \(E\) coincides with the closure of a control set \(D\) with nonvoid interior.
scientific article; zbMATH DE number 1901422