In this paper we study a variational problem derived from a computer vision application: video camera calibration with smoothing constraint. By video camera calibration we mean to estimate the location, orientation and lens zoom-setting of the camera for each video frame taking into account image visible features. To simplify the problem we assume that the camera is mounted on a tripod, in such case, for each frame captured at time t , the calibration is provided by 3 parameters : (1) P(t) (PAN) which represents the tripod vertical axis rotation, (2) T (t) (TILT) which represents the tripod horizontal axis rotation and (3) Z(t) (CAMERA ZOOM) the camera lens zoom setting. The calibration function t → u(t) = (P(t),T (t),Z(t)) is obtained as the minima of an energy function I[u]. In this paper we study the existence of minima of such energy function as well as the solutions of the associated Euler-Lagrange equations