Nonlinear bang-bang principle in \({\mathbb{R}}^ n\)
In the paper some interrelations between the sets A and B of all solutions of the Cauchy problems (1) \(\dot x\in F(t,x)\), \(x(0)=0\), \(t\in T\subset [0,1]\), \(x\in {\mathbb{R}}^ n\), (2) \(\dot x\in extr.F(t,x)\), \(x(0)=0\) are studied. It is assumed that F(\(\cdot,\cdot)\) is measurable in t, lipschitzian in x, has compact and convex values and that the function \(t\to \sup \{\| x\|:\) \(x\in F(t,0)\}\) is measurable and bounded. It is proved that the set B is dense in the set A. In the proof the Baire category theorem is used.
scientific article; zbMATH DE number 4212492
scientific article