Stabilization of a class of nonlinear systems by a smooth feedback control.

*(English)*Zbl 0569.93056The local stabilizability problem is considered for systems of the form \(\dot x=f(x)+ub\) with equilibrium point \(x_ e=0\), where u is a scalar control and \(b\in R^ n\) constant. If the uncontrollable part of the linearized system has only eigenvalues with real part \(\leq 0\) and \(r\geq 1\) eigenvalues on the imaginary axis the stabilizability question cannot be decided by linear analysis only. The author applies results from center manifold theory to study this problem via an associated system of reduced order r. General stabilizability criteria are not obtained but a sufficient criterion is derived for the special case where \(r=2\) and the linearized system has a pair of purely imaginary uncontrollable eigenvalues. Several interesting examples are discussed and, in particular, a simple two-dimensional system is presented which is locally and globally controllable but not stabilizable by smooth state feedback.

Reviewer: D.Hinrichsen

##### MSC:

93D15 | Stabilization of systems by feedback |

37C75 | Stability theory for smooth dynamical systems |

93C10 | Nonlinear systems in control theory |

93D99 | Stability of control systems |

93B05 | Controllability |

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

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##### References:

[1] | Aeyels, D., Local and global controllability for nonlinear systems, Systems control lett., 5, 19-26, (1984) · Zbl 0552.93009 |

[2] | Brockett, R.W., Asymptotic stability and feedback stabilization, differential geometric control theory, () · Zbl 0528.93051 |

[3] | Carr, J., Applications of centre manifold theory, (1981), Springer New York · Zbl 0464.58001 |

[4] | Takens, F., Singularities of vector fields, Publ. math. IHES, 43, 47-100, (1974) · Zbl 0279.58009 |

[5] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector. fields, (), 123-156 |

[6] | Sussmann, H.J., Subanalytic sets and feedback control, J. differential equations, 31, 1, 31-42, (1979) · Zbl 0407.93010 |

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