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On the Kalman problem. (English. Russian original) Zbl 0713.93044

Sib. Math. J. 29, No. 3, 333-341 (1988); translation from Sib. Mat. Zh. 29, No. 3(169), 3-11 (1988).
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.

MSC:

93D20 Asymptotic stability in control theory
34D20 Stability of solutions to ordinary differential equations
93C05 Linear systems in control theory
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