Optimal Boundary Control of 2x2 Linear Hyperbolic PDEs
  The present paper develops an optimal linear quadratic boundary controller for $2\times2$ linear hyperbolic partial differential equations (PDEs) with actuation on only one end of the domain. First-order necessary conditions for optimality is derived via weak variations and an optimal controller in state-feedback form is presented. The linear quadratic regulator (LQR) controller is calculated from differential algebraic Riccati equations. Numerical examples are performed to show the use of the proposed method. 
Agus Hasan, Lars Imsland, Ivan Ivanov, Snezhana Kostova, Boryana   Bogdanova
math.OC