  We show that it is possible to independently control both the trajectory and
the maximum amplitude along the trajectory of a paraxial accelerating beam.
This is accomplished by carefully engineering both the amplitude and the phase
of the beam on the input plane. Furthermore, we show that the width of an
accelerating beam is related only on the curvature of the trajectory.
Therefore, we are able to produce beams with predefined beam widths and
amplitudes. These results are useful in applications where precise beam control
is important. In addition we consider radially symmetric abruptly autofocusing
beams. We identify the important parameters that affect the focal
characteristics. Consequently, we can design autofocusing beams with optimized
parameters (such as sharper focus and higher intensity contrast). In all our
calculations the resulting formulas are presented in an elegant and practical
form in direct connection with the geometric properties of the trajectory.
Finally we discuss methods that can be utilized to experimentally realize such
optical waves.
