Stabilization of Integral Delay Equations by solving Fredholm equations

The paper proves that, if kernels f and g are chosen so that I1=I2=I3=0 on the specified intervals, then the closed-loop characteristic equation collapses to 1 = a e^{-τ0 s}, yielding exponential stability for x and U (Theorem 1) . The candidate solution gives a consistent alternative time-domain derivation via a causal Bezout convolution identity that reduces the closed-loop to x(t)=a x(t−τ0), then establishes exponential decay and bounds in the Cpw-norms. The only substantive omission in the candidate solution is the paper’s controllability/existence hypothesis (Assumption 1) required to guarantee that such f,g exist (Lemma 3) . Aside from that, both arguments align on the main stability conclusion.