Boundary state feedback exponential stabilization for a onedimensional wave equation with velocity recirculation Abstract   In this paper, we consider boundary state feedback stabilization of a onedimensional wave equation with indomain feedbackrecirculation of an intermediate point velocity. We firstly construct an auxiliary control system which has a nonlocal term of the displacement at the same intermediate point. Then by choosing a wellknown exponentially stable wave equation as its target system, we find one backstepping transformation from which a state feedback law for this auxiliary system is proposed. Finally, taking the resulting closedloop of the auxiliary system as a new target system, we obtain another backstepping transformation from which a boundary state feedback controller for the original system is designed. By the equivalence of three systems, the closedloop of original system is proved to be wellposed and exponentially stable. Some numerical simulations are presented to validate the theoretical results.

Adaptive output regulator for wave PDEs with unknown harmonic disturbance Abstract   This paper focuses on developing boundary controller for a wave PDE with two disturbances;     (  i  )     spatially varying and timeinvariant indomain disturbance,     (  i  i  )     timevarying sinusoidal boundary disturbance. We consider the case that the sensor is collocated with the actuator and harmonic disturbance is anticollocated with the sensoractuator. We aim to regulate the rate of change of the opposite boundary, relative to the actuator, around a given constant reference. Towards that aim, we reformulate the opposite boundary equation as a linear system with known parameters, the sinusoidal disturbances and simultaneous inputoutput delays. In this way, the control input and sensor become collocated with the disturbance. Then, we parametrize the disturbance and represent the delays as a transport PDE. We develop an observerbased adaptive boundary controller for the ODExe2x80x93PDE cascade system. We prove that all signals are bounded and the rate of change of the opposite boundary tracks a given constant reference. The performance of the controller is demonstrated in a numerical simulation.