Marcum Q-functions and explicit kernels for stabilization of 2×2 linear hyperbolic systems with constant coefficients

Abstract

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 -functions of the first order. The Marcum functions are common in certain applications in communications but have not appeared previously in control design problems. The dependence of the explicit solutions with respect to system parameters is analyzed through several examples, including the stabilization of a constant equilibrium for a quasi-linear plant.