##
**Phase portraits of planar control systems.**
*(English)*
Zbl 0858.93037

Linear, single input single output control systems in the plane of the form
\[
dx/dt=Ax+bu, \qquad y=cx\tag{\(*\)}
\]
are considered. They are subject to a saturated output feedback \(u=\varphi(y)\). The paper investigates dynamical systems arising as systems \((*)\) with closed feedback loop, and provides a qualitative characterization of phase portraits of these systems under the assumption that the origin is an isolated asymptotically stable equilibrium point. This characterization is made in terms of two parameters of the systems which are the trace and the determinant of the matrix \(A\). The results obtained in the paper establish possible shapes of phase portraits depending on the system parameters. In particular, it is proved that the systems may have two types of closed curve invariant sets: these are either unstable single closed orbits or they consist of a pair of heteroclinic orbits joining two saddle points.

Reviewer: K.Tchoń (Wrocław)

### MSC:

93C15 | Control/observation systems governed by ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |

34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

### Keywords:

control systems; plane; saturated output feedback; phase portraits; closed orbits; heteroclinic orbits
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\textit{J. Llibre} and \textit{J. Sotomayor}, Nonlinear Anal., Theory Methods Appl. 27, No. 10, 1177--1197 (1996; Zbl 0858.93037)

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### References:

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[3] | Alvarez, J.; Suárez, R.; Alvarez, J., (Planar Linear Systems with Single Saturated Feedback (1992), University Autonoma Metrop.: University Autonoma Metrop. New York), (preprint) |

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