Stabilizing Integral Delay Dynamics and Hyperbolic Systems using a Fredholm Transformation

In this paper, we design a stabilizing state-feedback control law for a system represented by a general class of integral delay equations. Under an appropriate spectral controllability assumption, an implementable control law is proposed. The approach is constructive and makes use of the well-known backstepping methodology. Due to the integral terms present in the original system, the proposed problem requires a Fredholm transform, which is not always invertible. The invertibility of this transformation is proved using an operator formulation. In particular, we show that this invertibility property is a consequence of spectral controllability. The existence of the kernels defining the Fredholm transform is proved similarly by showing that they satisfy an invertible integral equation. Some test case simulations complete the paper.