Optimal Control of Underdamped Systems: An Analytic Approach

Area of Science:

• Physics
• Physical Chemistry
• Statistical Mechanics

Background:

• Optimal control theory seeks to minimize costs in system transitions.
• Stochastic systems present challenges, especially in underdamped dynamics relevant to nanoscale electronics.
• Existing methods often focus on overdamped systems, limiting applicability.

Purpose of the Study:

• Develop analytic techniques for optimal control of stochastic underdamped dynamics.
• Minimize thermodynamic cost during finite-time transitions.
• Address challenges in nanoscale electronic component design.

Main Methods:

• Applied optimal control theory to underdamped stochastic dynamics.
• Utilized Kullback-Leibler divergence and mean entropy production as cost functions.
• Developed an infinite-dimensional Poincaré-Lindstedt perturbation theory for Maxwell-Boltzmann distributions.
• Solved Lyapunov equations for Gaussian state transitions.

Main Results:

• Derived optimal protocols for minimum thermodynamic cost in underdamped systems.
• Showed optimal protocols satisfy Lyapunov equations for Gaussian states.
• Introduced a novel perturbation theory improving standard multiscale expansions.
• Enabled explicit computation of momentum cumulants for underdamped dynamics.

Conclusions:

• The developed analytic techniques provide a robust framework for optimal control of underdamped stochastic systems.
• Results offer insights into thermodynamic costs and inertial effects in nanoscale systems.
• The new perturbation theory advances the study of non-equilibrium statistical mechanics.