Well-Posedness and Stability of Infinite-Dimensional Systems Under Monotone Feedback · arXivDesk
2503.16092Mar 20, 202540 pages, 1 figure. Submitted. Version 3: Minor changes in Section 4. Version 2: The stability results in Section 4 were generalised to system nodes which are impedance passive but not necessarily well-posed
Well-Posedness and Stability of Infinite-Dimensional Systems Under Monotone Feedback
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
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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.