During guidance and control procedures of orbiting spacecraft, the respect of positioning and space constraints is decisive for successful missions achievement. The development of algorithms capable of fulfilling these constraints is directly related to how precisely the spacecraft trajectories are known. Since accuracy is essential for these procedures, the prevention and estimation of errors arising from approximations and numerical computations become critical. In this context, we consider solving linear ordinary differential equations via rigorous polynomial approximations in Chebyshev series. These are polynomials together with an error bound accounting for both approximation and rounding errors. Our method allows for the computation of validated approximations of the transition matrices describing the evolution of spacecraft trajectories. The proposed approach is employed in the following applications: first, we consider the linearized impulsive rendezvous framework, demonstrating how to use rigorous polynomials approximations to provide a validated propagation of the relative dynamics between spacecraft; this is then exploited for the hovering phases of the spacecraft rendezvous, where we conceive a validated model predictive control based on semi-definite programs. Finally, we propose a semi-analytical transition matrix for a simplified model of geostationary station keeping, linearizing the spacecraft dynamics which take into account the J2 Earth oblateness effect.