Existence of schlicht integral manifolds for ordinary differential equations (Q1842027)

Consider a coupled pair of differential equations (1) \(\dot x= g(t, x, y)\), \(\dot y= h(t, x, y)\) \((x\in \mathbb{R}^{n_ 1}, y\in \mathbb{R}^{n_ 2}, n_ 1+ n_ 2= n)\), assume that each solution \((x(t), y(t))\) defined at some \(t= t_ 0\) can be extended to all \(t\in \mathbb{R}\). A subset of the \((x,y)\)-space or of the \((t, x, y)\)-space is called schlicht if it is the graph of a function \(y= s(x)\), resp. \(y= S(t, x)\); i.e., ``schlicht'' indicates uniqueness of \(y\). The author considers an integral manifold \(M(t)\) generated by solutions of (1) which start from a schlicht initial manifold \(M(t_ 0)\), i.e., with initial values satisfying \(y(t_ 0)= s(x(t_ 0))\); he presents conditions which guarantee that \(M(t)\) is schlicht for all \(t\) in some interval \([t_ 0, T]\), \(T\leq \infty\). Among others, this result has an interesting application in a new approach to the construction of observers in nonlinear control theory presented recently by Knobloch and Flockerzi (see [\textit{H. W. Knobloch}, \textit{A. Isidori} and \textit{D. Flockerzi}, Topics in control theory. DMV Seminar Vol. 22, Basel (1993; Zbl 0789.93073)], p. 129).