Total variation diminishing schemes in optimal control of scalar conservation laws

Optimal control problems subject to a nonlinear scalar conservation law, the state system, are studied. Such problems are challenging at both the continuous level and the discrete level since the control-to-state operator poses difficulties such as nondifferentiability. Therefore, discretization of the control problem has to be designed with care to provide stability and convergence. Here, the discretize-then-optimize approach is pursued, and the state is removed by the solution of the underlying state system, thus providing a reduced control problem. An adjoint calculus is then applied for computing the reduced gradient in a gradient-related descent scheme for solving the optimization problem. The time discretization of the underlying state system relies on total variation diminishing Runge–Kutta (TVD-RK) schemes, which guarantee stability, best possible order and convergence of the discrete adjoint to its continuous counterpart. While interesting in its own right, it also influences the quality and accuracy of the numerical reduced gradient and, thus, the accuracy of the numerical solution. In view of these demands, it is proven that providing a state scheme that is a strongly stable TVD-RK method is enough to ensure stability of the discrete adjoint state. It is also shown that requiring strong stability for both the discrete state and adjoint is too strong, confining one to a first-order method, regardless of the number of stages employed in the TVD-RK scheme. Given such a discretization, we further study order conditions for the discrete adjoint such that the numerical approximation is of the best possible order. Also, convergence of the discrete adjoint state towards its continuous counterpart is studied. In particular, it is shown that such a convergence result hinges on a regularity assumption at final time for a tracking-type objective in the control problem. Finally, numerical experiments validate our theoretical findings.