Parametric identification problem with a regularizer in the form of the total variation of the control

We consider a linear dynamical system, for which we need to reconstruct the control input on the basis of a noisy output. We form the corresponding family of parametric optimal control problems in which the performance criterion contains terms corresponding to the problem regularization and clearing the output signal from speckle noises. The weight coefficient multiplying the term used for noise filtration plays the role of a parameter in the family of problems. We prove a theorem that describes the properties of solutions of parametric problems in a neighborhood of a regular point, analyze the differential properties of solutions of that problem, and derive formulas for the computation of derivatives of the optimal trajectory and the optimal control with respect to a parameter. We suggest a simple method for constructing approximate solutions of perturbed optimal control problems. These results permit one to control the performance of the reconstruction of the control in the original identification problem. An illustrative example is considered.