Optimal control of scalar one-dimensional conservation laws

This paper presents an optimal control theory for scalar one-dimensional nonlinear hyperbolic partial differential equations also called conservation laws. The solution of these equations may develop discontinuities known as shock waves that forbid the use of classical variational techniques. This paper proposes to compute the first variation of the dynamics based on its weak formulation and gives an explicit formula of its solution. Adjoint calculus is then used to evaluate gradients of cost functionals that may contain the shock locations. An application to the Burgers equation is given as an illustration