## Abstract

We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.

Original language | English |
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Pages (from-to) | 327-360 |

Number of pages | 34 |

Journal | Mathematics of Control Signals and Systems |

Volume | 32 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 2020 |

## Keywords

- Absolute stability
- Almost periodic functions
- Circle criterion
- Incremental input-to-state stability
- Infinite-dimensional systems
- Lur’e systems
- Small gain

## ASJC Scopus subject areas

- Control and Systems Engineering
- Signal Processing
- Control and Optimization
- Applied Mathematics