Minimal conditions for dynamical stability refer to the necessary criteria that system parameters must meet to ensure its response to disturbances remains bounded over time. Lyapunov and Kapitza stability criteria are classical examples that have led to major technological advances and fundamental insights in physics. Here, we establish new minimal stability conditions for systems whose potential energy curvature varies periodically in time, extending Kapitza stabilization to what we term optimal dynamical stabilization. Using optimal control theory, we determine the minimal time-periodic stiffness required to stabilize a linear mass-spring system and validate our predictions with model experiments. When the potential curvature alternates between negative and positive values, the set of modulation functions ensuring optimal stability becomes discrete and is governed by eigenvalue problems mathematically analogous to those describing quantum bound states in one-dimensional potential wells. These findings deepen our understanding of dynamical systems and lay the foundation for promising new passive stabilization techniques in applied physics.

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• Control & applications of chaos
• Lagrangian dynamics
• Floquet systems
• Hamiltonian systems