On null controllability of linear systems with recurrent coefficients and constrained controls (Q1188333)

For a control system \(\dot x=A(t)+B(t)u(t)\) in \(R^ n\), with \(u(t)\) restricted to a convex compactum \(Q\subset\mathbb{R}^ m\) containing zero and \(\xi_ 0=(A,B)\) uniformly recurrent \((\xi\)'s \(t\)-shift \(\tau_ t(\xi)\) is \(\varepsilon\)-close to \(\xi\) for a \(t\) from every \(T\)-long interval, \(T=T(\varepsilon))\), the following local and global null controllability results are proved: (i) if for a certain \(\xi\in E=\{\tau_ t(A,B):t\in R\}\) the \(\xi\)- system (obtained from the initial one by replacing \(\xi_ 0\) by \(\xi)\) is locally null controllable, then the family of all \(\xi\)-systems is uniformly null controllable, i.e. there exist a neighborhood \(V\) of the origin in \(R^ n\) and a \(T>0\) such that every \(\xi\)-system can be steered from \(V\) to the origin in time \(T\); (ii) if the above uniform null controllability takes place, and the spectrum \(\Sigma\) of the homogeneous \(\xi\)-systems lies in \((-\infty,0]\), then the \(\xi\)-system is globally null controllable for \(\mu\)-almost all \(\xi\in E\) where \(\mu\) is an invariant measure on \(E\). The proof of (ii) is based on an ergodic theoretic result of \textit{R. J. Sacker} and \textit{G. R. Sell} [J. Differ. Equations 27, 320-358 (1978; Zbl 0372.34027)]. A partial conversion of (ii) is formulated.