## Abstract

We study existence of solutions, and in particular well-posedness, for a class of inhomogeneous, nonlinear partial differential equations (PDE's). The main idea is to use system theory to write the nonlinear PDE as a well-posed infinite-dimensional linear system interconnected with a static nonlinearity. By a simple example, it is shown that in general well-posedness of the closed-loop system is not guaranteed. We show that well-posedness of the closed-loop system is guaranteed for linear systems whose input to output map is coercive for small times interconnected to monotone nonlinearities. This work generalizes the results presented in [1], where only globally Lipschitz continuous nonlinearities were considered. Furthermore, it is shown that a general class of linear port-Hamiltonian systems satisfies the conditions asked on the open-loop system. The result is applied to show well-posedness of a system consisting of a vibrating string with nonlinear damping at the boundary.

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

Pages (from-to) | 19-25 |

Number of pages | 7 |

Journal | Systems and control letters |

Volume | 128 |

Early online date | 6 May 2019 |

DOIs | |

Publication status | Published - 1 Jun 2019 |

## Keywords

- Boundary feedback
- Nonlinear damping
- Nonlinear feedback
- Passive infinite-dimensional systems
- port-Hamiltonian systems
- Vibrating string
- Well-posedness