  We explore linear control of the one-dimensional non-linear
Kardar--Parisi--Zhang (KPZ) equation with the goal to understand the effects
the control process has on the dynamics and on the stationary state of the
resulting stochastic growth kinetics. In linear control, the intrinsic
non-linearity of the system is maintained at all times. In our protocol, the
control is applied to only a small number $n_c$ of Fourier modes. The
stationary-state roughness is obtained analytically in the small-$n_c$ regime
with weak non-linear coupling wherein the controlled growth process is found to
result in Edwards--Wilkinson dynamics. Furthermore, when the non-linear KPZ
coupling is strong, we discern a regime where the controlled dynamics shows
scaling in accordance to the KPZ universality class. We perform a detailed
numerical analysis to investigate the controlled dynamics subject to weak as
well as strong non-linearity. A first-order perturbation theory calculation
supports the simulation results in the weak non-linear regime. For strong
non-linearity, we find a temporal crossover between KPZ and dispersive growth
regimes, with the crossover time scaling with the number $n_c$ of controlled
Fourier modes. We observe that the height distribution is positively skewed,
indicating that as a consequence of the linear control, the surface morphology
displays fewer and smaller hills than in the uncontrolled growth process, and
that the inherent size-dependent stationary-state roughness provides an upper
limit for the roughness of the controlled system.
