Boundary control design for linear 1-D balance laws in the presence of in-domain disturbances

Boundary feedback control design for linear scalar conservation laws in the presence of exogenous disturbances is studied. The closed-loop system is rewritten as an abstract dynamical system. Sufficient conditions in the form of dissipation functional inequalities are given to establish the existence of a bound on the mathcal{L}{2} norm of the state with respect to energy bounded in-domain disturbances. A systematic design algorithm based on semidefinite programming is provided to design a stabilizing control gain minimizing the effect of the exogenous input on the closed-loop response. The effectiveness of the approach is shown in a numerical example.