Well-Posedness and Stability of Infinite-Dimensional Systems Under Monotone Feedback
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Hastir, Anthony Paunonen, Lassi |
| author_facet | Hastir, Anthony Paunonen, Lassi |
| contents | We study the well-posedness and stability of an impedance passive infinite-dimensional linear system under nonlinear feedback of the form $u(t)=ϕ(v(t)-y(t))$, where $ϕ$ is a monotone function. Our first main result introduces conditions guaranteeing the existence of classical and generalised solutions in a situation where the original linear system is well-posed. In the absence of the external input $v$ we establish the existence of strong and generalised solutions under strictly weaker conditions. Finally, we introduce conditions guaranteeing that the origin is a globally asymptotically stable equilibrium point of the closed-loop system. Motivated by the analysis of partial differential equations with nonlinear boundary conditions, we use our results to investigate the well-posedness and stablility of abstract boundary control systems, port-Hamiltonian systems, a Timoshenko beam model, and a two-dimensional boundary controlled heat equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-Posedness and Stability of Infinite-Dimensional Systems Under Monotone Feedback Hastir, Anthony Paunonen, Lassi Optimization and Control Analysis of PDEs Functional Analysis 34G20, 47H20, 93B52, 93D20 (47H06, 35B35, 35K05) We study the well-posedness and stability of an impedance passive infinite-dimensional linear system under nonlinear feedback of the form $u(t)=ϕ(v(t)-y(t))$, where $ϕ$ is a monotone function. Our first main result introduces conditions guaranteeing the existence of classical and generalised solutions in a situation where the original linear system is well-posed. In the absence of the external input $v$ we establish the existence of strong and generalised solutions under strictly weaker conditions. Finally, we introduce conditions guaranteeing that the origin is a globally asymptotically stable equilibrium point of the closed-loop system. Motivated by the analysis of partial differential equations with nonlinear boundary conditions, we use our results to investigate the well-posedness and stablility of abstract boundary control systems, port-Hamiltonian systems, a Timoshenko beam model, and a two-dimensional boundary controlled heat equation. |
| title | Well-Posedness and Stability of Infinite-Dimensional Systems Under Monotone Feedback |
| topic | Optimization and Control Analysis of PDEs Functional Analysis 34G20, 47H20, 93B52, 93D20 (47H06, 35B35, 35K05) |
| url | https://arxiv.org/abs/2503.16092 |