Marcum Q-functions and explicit feedback laws for stabilization of constant coefficient 2 × 2 linear hyperbolic systems

We find the exact analytical solution to a Goursat PDE system governing the kernels of a backstepping-based boundary control law that stabilizes a constant-coefficient 2 × 2 system of first-order hyperbolic linear PDEs. The solution to the Goursat system is related to the solution of a simpler, explicitly solvable Goursat system through a suitable infinite series of powers of partial derivatives which is summed explicitly in terms of special functions, including Bessel functions and the generalized Marcum Q-functions of the first order. The Marcum functions are common in certain applications in communications but have not appeared previously in control design problems.