On the Kalman problem (Q750389)

Consider a system (1) \(dx/dt=Ax+b\phi (\sigma)\), \(\sigma =c^*x\), where A is a square matrix, b and c are column vectors, and \(\phi\) is a continuous scalar function. Assume that for all \(\mu\in (\alpha,\beta)\) system (1) with \(\phi (\sigma)=\mu \sigma\) is asymptotically stable. Strengthening the assumptions in Aizerman's problem, R. E. Kalman conjectured that if \(\phi '(\sigma)\in (\alpha,\beta)\) for all \(\sigma\) then the system is globally asymptotically stable. In this interesting paper first Kalman's conjecture is proved if the dimension n of the system is 3. Secondly, it is shown using an explicit counterexample that if \(n\geq 4\) then systems exist which satisfy Kalman's condition and still have a nontrivial periodic solution.