Optimal dynamical stabilization
Arnaud Lazarus1, Emmanuel Trélat2
1Massachusetts Institute of Technology, Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, UMR7190, Paris, France and , Department of Mathematics, Cambridge, Massachusetts 02139, USA.
Physical Review. E
|April 18, 2026
Summary
New minimal stability conditions, termed optimal dynamical stabilization, were found for systems with periodically varying potential energy. This extends Kapitza stabilization and offers new passive stabilization techniques.
Area of Science:
- Physics
- Dynamical Systems
- Control Theory
Background:
- Dynamical stability ensures bounded system responses to disturbances.
- Classical criteria like Lyapunov and Kapitza stability are crucial in physics and technology.
- Understanding minimal stability conditions is key for system design.
Purpose of the Study:
- To establish new minimal conditions for dynamical stability in systems with time-varying potential energy.
- To extend the concept of Kapitza stabilization to 'optimal dynamical stabilization'.
- To determine the minimal time-periodic stiffness for stabilizing a linear mass-spring system.
Main Methods:
- Utilizing optimal control theory to find minimal stabilization parameters.
- Applying mathematical analysis to systems with periodically alternating potential energy curvature.
- Validating theoretical predictions through model experiments.
Main Results:
- Identified new minimal conditions for dynamical stability in periodically modulated systems.
- Determined the minimal time-periodic stiffness for a linear mass-spring system.
- Found that alternating potential curvature leads to discrete modulation functions governed by eigenvalue problems analogous to quantum mechanics.
Conclusions:
- The study introduces 'optimal dynamical stabilization' for systems with time-varying potential energy.
- Findings provide a deeper understanding of dynamical systems and their stability.
- The research lays groundwork for novel passive stabilization techniques in applied physics.
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