## Abstract

Recently, a constructive approach to the design of finite-dimensional observer-based controller has been proposed for parabolic partial differential equations (PDEs). This paper extends it to hyperbolic PDEs. Namely, we design a finite-dimensional, output- feedback, boundary controller for a wave equation with in-domain viscous friction. The control-free system is unstable for any friction coefficient due to an external force. Our approach is based on modal decomposition: an observer-based controller is designed for a finite-dimensional projection of the wave equation on <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> eigenfunctions (modes) of the Sturm-Liouville operator. The danger of this approach is the “spillover” effect: such a controller may have a deteriorating effect on the stability of the unconsidered modes and cause instability of the full system. Our main contribution is an appropriate Lyapunov-based analysis leading to linear matrix inequalities (LMIs) that allow one to find a controller gain and number of modes, <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, guaranteeing that the “spillover” effect does not occur. An important merit of the derived LMIs is that their complexity does not change when <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> grows. Moreover, we show that appropriate <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> always exists and, if the LMIs are feasible for some <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, they remain so for <inline-formula><tex-math notation="LaTeX">$N+1$</tex-math></inline-formula>.

Original language | English |
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Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | IEEE Transactions on Automatic Control |

DOIs | |

State | Accepted/In press - 2023 |

## Keywords

- Aerospace electronics
- Damping
- Friction
- Linear matrix inequalities
- Observers
- Propagation
- Reduced order systems