Communications in Mathematical Sciences

Volume 13 (2015)

Number 1

Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs

Pages: 15 – 48



Michael Herty (Department of Mathematics, RWTH Aachen University, Aachen, Germany)

Alexander Kurganov (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)

Dmitry Kurochkin (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)


We develop a numerical method for the solution to linear adjoint equations arising, for example, in optimization problems governed by hyperbolic systems of nonlinear conservation and balance laws in one space dimension. Formally, the solution requires one to numerically solve the hyperbolic system forward in time and a corresponding linear adjoint system backward in time. Numerical results for the control problem constrained by either the Euler equations of gas dynamics or isothermal gas dynamics equations are presented. Both smooth and discontinuous prescribed terminal states are considered.


PDE-constrained optimization problems, hyperbolic systems of conservation and balance laws, linear adjoint system, least-square cost functional, central-upwind finite-volume scheme, upwind finite-difference scheme

2010 Mathematics Subject Classification

49K20, 49M05, 49M29, 65M06, 65M08, 65M20

Published 16 July 2014