Control and stabilisation of strongly coupled hyperbolic systems with kinetic boundary conditions and implicit memory effects

In this paper, we investigate a coupled system of hyperbolic equations with control applied to one of the two parameters. A key feature of this work is the integration of a kernel modelled as a functional that implicitly incorporates the system's history. The memory term is represented by an integral involving a fractional operator with exponent ℓ, where the amplitude of the functional is explicitly linked to this exponent. The interaction between the amplitude ϱ and the exponent ℓ plays a crucial role in the dynamic behaviour and stability properties of the system. We demonstrate that the polynomial stability depends on the interaction between the memory kernel, the fractional operator's exponent ℓ, and the amplitude ϱ, while also studying the system's optimal controllability. Under appropriate assumptions, we establish that the solutions decay at the rate t−ϱ1−ℓ, where the decay is influenced by both the amplitude and the infinite memory. Furthermore, detailed spectral analysis provides asymptotic expressions for the eigenvalues, confirming the optimality of the obtained decay rates.