Observability and observer design for a class of hyperbolic PDEs with van de Pol type boundary conditions

This paper focuses on observability and observer design for nonlinear complex dynamical systems described by a class of hyperbolic partial differential equations (PDEs) with nonlinear van de Pol type boundary conditions. The systems exhibit complex dynamics due to its imbalance of energy flows. Both the exact observability and approximate observability of the systems with different boundary output measurements are shown by using methods of characteristics and boundary nonlinear reflections. Motivated by the approximate observability of the systems, a PDE state observer by using the boundary displacement measurement only is designed, and a sufficient condition to guarantee the estimation error systems to be exponentially stable is given. Theoretical results are proved rigorously, with some numerical simulations performed to validate the effect of the proposed observer. (c) 2023 Elsevier B.V. All rights reserved.