## Abstract

Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the one dimensional (1D) heat equation, under the assumption that one of the observation or control operators is bounded. This article suggests a constructive method for such controllers for 1D parabolic partial differential equations (PDEs) with both observation and control operators being unbounded. We consider the Kuramoto-Sivashinsky equation under either boundary or in-domain point measurement and boundary actuation in the presence of disturbances in the PDE and measurement. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension N may be larger. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in a linear matrix inequality (LMI), for input-to-state stabilization (ISS) and guaranteed L2-gain, whose elements and dimension depend on N. The value of N and the decay rate are obtained from the LMI. We prove that the LMI is always feasible provided N and the L2 or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some N implies its feasibility for N+1. Numerical examples demonstrate the efficiency of the method.

Original language | English |
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Pages (from-to) | 5570-5577 |

Number of pages | 8 |

Journal | IEEE Transactions on Automatic Control |

Volume | 67 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2022 |

### Funding

Funders | Funder number |
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Israel Science Foundation | 673/19 |

## Keywords

- Boundary control
- LMI
- modal decomposition
- observer-based control
- parabolic PDEs

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