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.

This paper presents bilateral control laws for one-dimensional(1-D) linear 2 × 2 hyperbolic first-order systems (with spatially varying coefficients). Bilateral control means there are two actuators at each end of the domain. This situation becomes more complex as the transport velocities are no longer constant, and this extension is nontrivial. By selecting the appropriate backstepping transformation and target system, the infinite-dimensional backstepping method is extended and a full-state feedback control law is given that ensures the closed-loop system converges to its zero equilibrium in finite time. The design of bilateral controllers enables a potential for fault-tolerant designs.