Reconstruction of controls in parabolic systems by Tikhonov's method with nonsmooth stabilizers

The problem of reconstructing a priori unknown controls in parabolic systems by the results of approximate a posteriori observations of the motion of this system is considered. It is proposed to solve this problem by Tikhonov's method with a stabilizer containing the total time variation of the varying control. The use of such a nondifferentiable stabilizer allows one to obtain in some cases more precise results than the approximation of the desired control in Lebesgue spaces. In particular, it becomes possible to establish the piecewise uniform convergence of regularized approximations, which can be used for the numerical reconstruction of the subtle structure of the desired control. Results of numerical experiments are presented.