With the advent of quantum technologies, control issues are becoming increasingly important. In this paper, we address control in phase space under a global constraint provided by a minimal energylike cost function and a local (in Fourier space) constraint resulting from a robustness criterion. The inverse-engineering Lagrangian formalism developed here generalizes the formalism commonly used to describe the superoscillation phenomenon. It is applied to both nondissipative and dissipative quantum mechanics, and extended to stochastic thermodynamics. Interestingly, our approach also allows one to improve sensing capabilities by appropriate control of the system.