Control of 2D scalar conservation laws in the presence of shocks

Abstract

Abstract. We analyze a model optimal control problem for a 2D scalar conservation law: The so-called inverse design problem, the goal being to identify the initial datum leading to a given final time configuration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and finite differences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The efficiency of the method is illustrated by numerical experiments.

Initial data identification in conservation laws and Hamilton–Jacobi equations

“…Here, the present result amounts to characterizing the terminal cost corresponding to given initial cost, see [16,Section 10.3] for further connections to optimal control problems. The present analytic results can also help in numerical investigations such as those in [2,10,26,27]. Sections 2 to 5 collect the analytic results, while all proofs are deferred to sections 6 to 9…”

A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws

“…Theorem 3.1 Let f (y), y(t, ⋅) ∈ C 3 (ℝ) and p(t, ⋅) ∈ C 4 (ℝ) . Then the adjoint WENO3 scheme (20) is adjoint consistent of order three, i.e., r j (t) = O(Δx 3 ) in (24).…”

“…In general, taking the adjoint as a decent direction may increase the complexity of the optimization process due to the production of additional discontinuities [5,23,24]. A careful choice of the initial guess u 0 can remedy this serious problem.…”

A numerical procedure and coupled system formulation for the adjoint approach in hyperbolic PDE-constrained optimization problems

“…One of our main motivations applying an optimal control strategy to general conservation laws is the possibility of study numerical inverse problems related to this type of systems. Example of them are the bottom detection in water-waves type system, see [29], or the tracking of initial conditions in the presence of shocks [5,25], to mention but a few.…”

“…Moreover, the present approach works even if the constraint PDE is non-conservative. In the literature, there exist several works dealing with minimization considering hyperbolic conservation laws, see [18,17,5,25,39] to mention but a few. However, to best of our knowledge, the present approach has not been reported previously and this incorporates in a natural way the non-conservative structure of the adjoint model.…”