On global controllability of linear time dependent control systems (Q809954)

The linear time-dependent system \(x'(t)=A(t)x(t)+B(t)u(t)\), \(t\in J=(\alpha,\beta)\subset {\mathbb{R}}\), is considered, where \(A,B\in L^ l_{loc}(J)\) and \(u\in U_ B=\{u\in L^ l_{loc}(J): Bu\in L^ l_{loc}(J)\}\). The properties of the function \(\tau (t)=\inf \{t'>t:(A,B)\) is \([t,t']\)- globally controllable from 0 by means of a set \(U\subset U_ B\}\) are emphasized. It is shown that if \(B\in L^ 2_{loc}(J)\), \(U=L^ 2_{loc}(J)\), then \(\tau\) is piecewise linear; more precisely it is piecewise constant, piecewise the identity.