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Maxwell-Bloch equations with a quadratic control about \(Ox_1\) axis. (English) Zbl 1085.34536

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 280-286 (2001).
The paper deals with a variant of the real Maxwell-Bloch equation, where a control is added to the first equation. The authors consider only feedback controls of the type \(u=-kx_2 x_3\), where \(k\) is a real constant. Thus, the aim is to characterize the dynamics of a system of ODEs dependent on a real parameter \(k\). It is shown that the system admits an infinite number of Hamilton-Poisson realizations. The stability of equilibria is analyzed as a function of the parameter \(k\). The system can be integrated via elliptic functions.
For the entire collection see [Zbl 0957.00038].

MSC:

34D20 Stability of solutions to ordinary differential equations
34A26 Geometric methods in ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
93B52 Feedback control
93D15 Stabilization of systems by feedback
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