For systems linear in control we consider problems of recovering the dynamics and control from a posteriori statistics of trajectory sampling and known estimates for the sampling error. An optimal control problem of minimizing an integral regularized functional of dynamics and statistics residuals is introduced. Optimal synthesis is used to construct controls and trajectories that approximate a solution of the inverse problem. A numerical approximation method based on the method of characteristics for the Hamilton-Jacobi-Bellman equation and on the notion of minimax/viscosity solution is developed. Sufficient conditions are obtained under which the proposed approximations converge to a normal solution of the inverse problem under a matched vanishing of the approximation parameters (bounds for the sampling error, the regularizing parameter, the grid step in the state variable, and the integration step). Results of the numerical solution of problems of identification and control and trajectory recovery are presented for a mechanical model of gravitation under given statistics of phase coordinate sampling