### Abstract

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.

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

Pages (from-to) | 617-647 |

Number of pages | 31 |

Journal | Journal of evolution equations |

Volume | 16 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 2016 |

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### Keywords

- Contraction semigroup
- Existence of solutions
- Output feedback
- Well-posed system

### Cite this

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*Journal of evolution equations*, vol. 16, no. 3, pp. 617-647. https://doi.org/10.1007/s00028-015-0315-1

**Feedback theory extended for proving generation of contraction semigroups.** / Kurula, Mikael; Zwart, Hans.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Feedback theory extended for proving generation of contraction semigroups

AU - Kurula, Mikael

AU - Zwart, Hans

PY - 2016/9/1

Y1 - 2016/9/1

N2 - Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.

AB - Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.

KW - Contraction semigroup

KW - Existence of solutions

KW - Output feedback

KW - Well-posed system

UR - http://www.scopus.com/inward/record.url?scp=84953255547&partnerID=8YFLogxK

U2 - 10.1007/s00028-015-0315-1

DO - 10.1007/s00028-015-0315-1

M3 - Article

VL - 16

SP - 617

EP - 647

JO - Journal of evolution equations

JF - Journal of evolution equations

SN - 1424-3199

IS - 3

ER -