  Quantum many-body systems are emerging as key elements in the quest for
quantum-based technologies and in the study of fundamental physics. In this
study, we address the challenge of achieving fast and high-fidelity evolutions
across quantum phase transitions, a crucial requirement for practical
applications. We introduce a control technique based on dynamical invariants
tailored to ensure adiabatic-like evolution within the lowest-energy subspace
of the many-body systems described by the transverse-field Ising and long-range
Kitaev models. By tuning the controllable parameter according to analytical
control results, we achieve high-fidelity evolutions operating close to the
speed limit. Remarkably, our approach leads to the breakdown of Kibble-Zurek
scaling laws, offering tunable and significantly improved time scaling
behavior. We provide detailed numerical simulations to illustrate our findings,
demonstrating scalability with the system size and robustness against noisy
controls and disorder, as well as its applicability to a non-integrable system.
