T. Liard, E. Zuazua. Inverse design for the one-dimensional Burgers equation. (2019)
Abstract. In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an optimal control problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in $L^2(R)$ norm. The two main contributions of this work are the following:
- We fully characterize the set of minimizers of the aforementioned optimal control problem
- A wave-front tracking method is implemented to construct numerically all of them
One of minimizers is the backward entropy solution, constructed using a backward-forward method.