Control reconstruction in hyperbolic systems

We consider an inverse dynamics problem which is to reconstruct a priori unknown distributed controls in a hyperbolic system given the results of approximate observations of the movements of this system. To solve this ill‐posed problem, we propose to use the Tikhonov’s method with a regularizer containing the sum of mean squared norm and the total variation over the time of an admissible control. Using such a regularizer lets one get, in a number of cases, better results than just approximating the control in question in Lebesgue spaces. In particular, along these lines we can establish pointwise and piecewise uniform convergence for regularized approximations, which opens up new opportunities for numerical reconstruction of the fine structure of the control. We give numerical modeling results.