Continuous Adjoint-Based Optimization of Hyperbolic Equations with Nonlinear Differential Equation Constraints on 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 multi-objective 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.