Given an n-dimensional, connected, analytic manifold M and analytic vector fields \(\{f_i\}_{i=0}^m\) on M consider the affine control process

Let \(t\rightarrow \xi _{x,\overline{u}}(t)\) be the solution curve satisfying the above differential equation with initial condition \(\xi (0) = x\) and \(\overline{u} = (u_1,u_2, \ldots ,u_m)\in W_T^m \equiv L^{\infty }([0,T],{\mathbb {R}}^m)\) is a control function. For a fixed \(x\in M\) and \(T>0\) consider the evaluation map \(E^x : W_T^m \rightarrow M : E^x(\overline{u}) = \xi _{x,\overline{u}}(T)\). Given a compact, semi-analytic set \(K\subset M\), we prove that, (Theorem 1.6), if the system satisfies SAP (strong accessibility property), at each \(x\in K\), then the set of control functions \(\overline{u}\in C^{\infty }([0,T],{\mathbb {R}}^m)\) at which the the map \(E^x\) is a submersion, (uniformly as x varies over K), is dense in the set \(C^\infty ([0,T],{\mathbb {R}}^m)\) of \(C^\infty \) control functions with the \(C^\infty \) topology. The proof of the uniform submersion theorem, uses a control theoretic notion of ‘detectability’ and is based on an important result, (Theorem 2.2), about existence and genericity of ‘universal detectors’.

给定一个Ñ维，连接，解析流形中号和分析矢量场\（\ {f_i \} _ {i = 0} ^ M \）上中号考虑仿射控制处理

设\(t\rightarrow \xi _{x,\overline{u}}(t)\)为满足上述微分方程的解曲线，初始条件为\(\xi (0) = x\)和\(\ overline{u} = (u_1,u_2, \ldots ,u_m)\in W_T^m \equiv L^{\infty }([0,T],{\mathbb {R}}^m)\)是一个控制功能。对于固定的\(x\in M\)和\(T>0\)考虑评估图\(E^x : W_T^m \rightarrow M : E^x(\overline{u}) = \xi _ {x,\overline{u}}(T)\)。给定一个紧凑的半解析集\(K\subset M\)，我们证明，（定理 1.6），如果系统满足 SAP（强可达性），在每个\(x\in K\)，那么控制功能集\(\overline{u}\in C^{\infty }([0,T],{\mathbb {R}}^m)\)其中地图\(E^x\)是一个淹没， （随着x在K 上的变化一致），在\(C^\infty \)控制函数的集合\(C^\infty ([0,T],{\mathbb {R}}^m)\)中是密集的与\(C^\infty \)拓扑。均匀淹没定理的证明使用了“可检测性”的控制理论概念，并基于一个重要的结果（定理 2.2），即关于“通用检测器”的存在性和通用性。