Adjoint optimization of one-dimensional hyperbolic equations with constrained periodic boundary conditions

This paper presents a continuous adjoint-based optimization theory for a general closed-loop transport hyperbolic model controlled via a periodic boundary control to minimize a cost functional. The periodic boundary control is subject to a nonlinear differential equation constraint, thus resulting in a coupling between the hyperbolic equation and the ordinary differential equation. Variational principles are used to derive the Pontryagin’s minimum principle for optimality that results in a dual adjoint system. A numerical optimization method is implemented using the adjoint-based second-order gradient method to solve for the optimal trajectory of the control. Numerical methods for solving the hyperbolic equation using an explicit scheme, wave splitting method and for solving the adjoint equation using an implicit scheme and a quasi-steady state method are described.