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The present manuscript considers the observer design problem for a class of scalar semi-linear hyperbolic partial differential equation (PDE) systems with a recycle loop through the boundary point. The design method of Kazantzis and Kravaris [(1998). Nonlinear observer design using Lyapunov's auxiliary theorem. Systems & Control Letters, 34(5), 241–247] developed for the nonlinear finite dimensional systems observer design is extended to semi-linear hyperbolic PDE systems. The observer design problem is tackled through a first-order associated PDE. Due to the existence of spatial partial derivative operator, Lyapunov's Auxiliary Theorem originated from finite-dimensional systems can be no longer applied to seek conditions ensuring solvability of the associated PDE. In this manuscript, a new theorem is formulated and proved to ensure the solvability. The solution for the associated PDE is locally analytic nonlinear coordinate transformation which provides foundation for observer realization and a series solution approach is developed. Simulation examples are presented to study the performance of the proposed observer. [ABSTRACT FROM AUTHOR]

This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation. [ABSTRACT FROM AUTHOR]