TY - JOUR

T1 - Well-posedness of infinite-dimensional linear systems with nonlinear feedback

AU - Hastir, Anthony

AU - Califano, Federico

AU - Zwart, Hans

PY - 2019/6

Y1 - 2019/6

N2 - 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.

AB - 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.

KW - Boundary feedback

KW - Nonlinear damping

KW - Nonlinear feedback

KW - Passive infinite-dimensional systems

KW - port-Hamiltonian systems

KW - Vibrating string

KW - Well-posedness

UR - http://www.scopus.com/inward/record.url?scp=85065042437&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2019.04.002

DO - 10.1016/j.sysconle.2019.04.002

M3 - Article

AN - SCOPUS:85065042437

VL - 128

SP - 19

EP - 25

JO - Systems and Control Letters

JF - Systems and Control Letters

SN - 0167-6911

ER -