Inverse design of one dimensional Burgers equation
We study the inverse design of one-dimensional Burgers equation which consists of identifying the set of initial data evolving to a given target at a final time. This leads to an ill-posed backward Cauchy problem. On one hand, the given target may be unreachable along forward entropic evolution or there exist multiple initial data leading to the same given target. The two main results are the follows A wave-front tracking method is implemented to identify randomly all the possible initial data yielding entropy solutions that coincide with a given target at time T. When the target function uT is unreachable, we fully characterize the set of initial data generating entropy solutions leading as close as possible to the given target uT in L2-norm.