## Abstract

We consider well-posed linear systems whose state trajectories satisfy ẋ = Ax + Bu, where u is the input and A is an essentially skew-adjoint and dissipative operator on the Hilbert space X. This means that the domains of A* and A are equal and A* + A = -Q, where Q ≥ 0 is bounded on X. The control operator B is possibly unbounded, but admissible and the observation operator of the system is B*. Such a description fits many wave and beam equations with colocated sensors and actuators, and it has been shown for many particular cases that the feedback u = -ky + v, with K > 0, stabilizes the system, strongly or even exponentially. Here, y is the output of the system and v is the new input. We show, by means of a counterexample, that if B is sufficiently unbounded, then such a feedback may be unsuitable: the closed-loop semigroup may even grow exponentially. (Our counterexample is a simple regular system with feedthrough operator zero.) However, we prove that if the original system is exactly controllable and observable and if k is sufficiently small, then the closed-loop system is exponentially stable.

Original language | English |
---|---|

Pages (from-to) | 273-297 |

Number of pages | 25 |

Journal | SIAM Journal on Control and Optimization |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - 2006 |

Externally published | Yes |

## Keywords

- Colocated sensors and actuators
- Exact controllability and observability
- Exponential stability
- Output feedback
- Positive-real transfer function
- Regular linear system
- Skew-adjoint operator
- Well-posed linear system