Optimal Control of the Euler Equations via Relaxation Approaches

Abstract

We discuss two linear relaxation approaches to the optimal control of nonlinear hyperbolic systems, in particular the control of Euler flows in gas dynamics. The first method is a relaxation system that is due to Jin and Xin [2], the second one is a Lattice-Boltzman approach [3], where we use one spatial dimension and five velocities (D1Q5 model). Both methods are incorporated in an adjoint based steepest-descent algorithm for the optimisation. Convincing numerical results are presented for both methods for an example with discontinuous solutions.We consider the optimal control of the one-dimensional Euler equations, which is of the formwhere u 0 is the control, u d is a given desired state and F : R n → R n is a nonlinear flux function, that in this case is given bywhere ρ is the density, m the momentum and E the energy of the considered gas. The velocity and the pressure of the gas can be derived by v = m/ρ and p = (γ−1)(E−1/2 ρ v), where γ denotes the specific heat ratio. Such kind of optimal control problems arise e.g. in gas dynamics and are difficult to solve mainly due to wave interactions that might occur in the solution of nonlinear systems of conservation laws. Relaxation MethodsRelaxation System: The first relaxation method that we are concerned with is the relaxation system that is proposed by Jin and Xin in [2]. The semi-linear approximation of the nonlinear hyperbolic equation ∂ t u + ∂ x F(u) = 0 with initial conditions u(x, 0) = u 0 is given bywhere ε > 0 is the relaxation rate and A = diag(a 1 , .