The current generation of ground-based coronagraphic instruments uses deformable mirrors to correct for phase errors and to improve contrast levels at small angular separations. Improving these techniques, several space and ground based instruments are currently developed using two deformable mirrors to correct for both phase and amplitude errors. However, as wavefront control techniques improve, more complex telescope pupil geometries (support structures, segmentation) will soon be a limiting factor for these next generation coronagraphic instruments. In this paper we discuss fundamental limits associated with wavefront control with deformable mirrors in high contrast coronagraph. We start with an analytic prescription of wavefront errors, along with their wavelength dependence, and propagate them through coronagraph models. We then consider a few wavefront control architectures, number of deformable mirrors and their placement in the optical train of the instrument, and algorithms that can be used to cancel the starlight scattered by these wavefront errors over a finite bandpass. For each configuration we derive the residual contrast as a function of bandwidth and of the properties of the incoming wavefront. This result has consequences when setting the wavefront requirements, along with the wavefront control architecture of future high contrast instrument both from the ground and from space. In particular we show that these limits can severely affect the effective Outer Working Angle that can be achieved by a given coronagraph instrument.