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Observer-based global output feedback controller design for an unstable wave PDE with nonlinear boundary condition. (English) Zbl 1485.93184

Summary: In this work, an observer-based output feedback controller for a one-dimensional wave equation with van der Pol type nonlinear boundary conditions is considered. The uncontrolled system with the energy injection and the cubic velocity nonlinearity reveals different kinds of dynamical behaviours, for example, chaotic acoustic vibration, period-doubling bifurcation, square wave and some other dynamical patterns. Firstly, an observer is designed, and next, we propose an observer-based output feedback controller using the backstepping approach. We prove well-posedness and globally asymptotic stability of the observer error and corresponding closed-loop system. To this purpose, pre-compactness of the trajectories is shown for the large-time analysis. The validity of the theoretical results is illustrated by some numerical simulations.

MSC:

93B52 Feedback control
93B53 Observers
93C20 Control/observation systems governed by partial differential equations
35L05 Wave equation
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