### Uittreksel

Originele taal-2 | Engels |
---|---|

Pagina's (van-tot) | 813-818 |

Tijdschrift | Differential Equations |

Volume | 36 |

Nummer van het tijdschrift | 6 |

DOI's | |

Status | Gepubliceerd - 2000 |

### Vingerafdruk

### Citeer dit

*Differential Equations*,

*36*(6), 813-818. https://doi.org/10.1007/BF02754404

}

*Differential Equations*, vol. 36, nr. 6, blz. 813-818. https://doi.org/10.1007/BF02754404

**A controlled closing theorem.** / Pavlov, A.; Fradkov, A.L.

Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic › peer review

TY - JOUR

T1 - A controlled closing theorem

AU - Pavlov, A.

AU - Fradkov, A.L.

PY - 2000

Y1 - 2000

N2 - In a number of problems in the theory of nonlinear oscillations and the theory of nonlinear control systems, one must investigate the behavior of solutions of differential systems under small changes of right-hand sides. In particular, the following problem is of interest: is it possible to transform a nearly periodic motion (for example, an almost periodic or recurrent motion) into a periodic motion with the use of small admissible changes of the right-hand side? The positive answer to this question was given in the so-called closing lemma proved by C. Pugh. However, in problems of control of oscillations, which have been intensively studied in the recent years, arbitrary variations of the right-hand sides are not allowed; only variations compatible with the actual capabilities of the control are admissible. Therefore, it is of interest to generalize the closing lemma to controlled systems. Some conditions of this type have been stated in the previous works by the authors. They essentially pertain to control theory and provide a criterion for the controllability of a nonlinear system near a recurrent trajectory by controls small in the uniform metric. In the present paper, we give a rigorous statement and a complete proof of this assertion (referred to as the controlled closing theorem).

AB - In a number of problems in the theory of nonlinear oscillations and the theory of nonlinear control systems, one must investigate the behavior of solutions of differential systems under small changes of right-hand sides. In particular, the following problem is of interest: is it possible to transform a nearly periodic motion (for example, an almost periodic or recurrent motion) into a periodic motion with the use of small admissible changes of the right-hand side? The positive answer to this question was given in the so-called closing lemma proved by C. Pugh. However, in problems of control of oscillations, which have been intensively studied in the recent years, arbitrary variations of the right-hand sides are not allowed; only variations compatible with the actual capabilities of the control are admissible. Therefore, it is of interest to generalize the closing lemma to controlled systems. Some conditions of this type have been stated in the previous works by the authors. They essentially pertain to control theory and provide a criterion for the controllability of a nonlinear system near a recurrent trajectory by controls small in the uniform metric. In the present paper, we give a rigorous statement and a complete proof of this assertion (referred to as the controlled closing theorem).

U2 - 10.1007/BF02754404

DO - 10.1007/BF02754404

M3 - Article

VL - 36

SP - 813

EP - 818

JO - Differential Equations

JF - Differential Equations

SN - 0012-2661

IS - 6

ER -