Vector method of design of sliding motion and simplex algorithms (Q1086538)

The problem considered is to find a partition \(A_ i\),\( i=1,...,m+1\), of \(\mathbb R^ m\) and a control \(w\) of the form \(w=w^ i(x,t)\), \((x,t)\in A_ i\), such that the motion of the system \(\dot x=f(x,t,w)\), \(x\in\mathbb R^{m}\), is stable with respect to a smooth manifold \(s(x,t)=0\). It is assumed that for some \(w^ i\) the derivative of \(s\) along the system equation has the form \(h(x,t)+H(x,t)u^ i\) where \(\det H\neq 0\). Two solutions are given.