Delay Boundary Stabilization of Coupled Linear Hyperbolic PDEs with Zero Characteristic Speed

In this research, we propose a novel control scheme to compensate for the effects of arbitrarily long input delays in heterodirectional hyperbolic partial differential equation systems with zero transport speed. Based on the delay phenomenon in the transport equation, the input delay is first transformed into a new transport equation, resulting in an equivalent system without delay. The controller is then designed using the backstepping method, in which the backstepping transformation consists of two classical second-type Volterra transformations and one affine-Volterra transformation. Unlike the Volterra transformation kernels, which are defined on triangular domains, the affine-Volterra transformation kernel is defined on a square domain. Proving the well-posedness of this kernel is the main challenge encountered in this work. Moreover, the presence of zero speed renders the invertibility of the affine-Volterra transformation less straightforward. With additional efforts, we demonstrate its invertibility. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed control scheme.