Optimal Control of Underdamped Systems: An Analytic Approach

Abstract

Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback-Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Linstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation.

74 pages, 8 figures

References in corpus (29)

• Optimal finite-time processes in stochastic thermodynamics
• Levitodynamics: Levitation and control of microscopic objects in vacuum
• Inertial effects of self-propelled particles: from active Brownian to active Langevin motion
• Command of active matter by topological defects and patterns
• Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control
• Finite-time Landauer principle
• Optimal protocols for minimal work processes in underdamped stochastic thermodynamics
• Fluctuation Relations for Diffusion Processes
• Efficiency of molecular motors at maximum power
• Information and thermodynamics: fast and precise approach to Landauer's bound in an underdamped micro-mechanical oscillator
• A geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems
• Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics
• Universal Bound on Energy Cost of Bit Reset in Finite Time
• Optimal finite-time bit erasure under full control
• Heating and Cooling are Fundamentally Asymmetric and Evolve Along Distinct Pathways
• Active matter under control: Insights from response theory
• Optimal transport and control of active drops
• Dynamics of information erasure and extension of Landauer's bound to fast processes
• A unified, geometric framework for nonequilibrium protocol optimization
• On how nanomechanical systems can minimize dissipation
• E. Schrödinger's 1931 paper "On the Reversal of the Laws of Nature" ["Über die Umkehrung der Naturgesetze",Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalische mathematische Klasse, 8 N9 144-153]
• Optimality of non-conservative driving for finite-time processes with discrete states
• Control of Active Brownian Particles: An exact solution
• Taming the time evolution in overdamped systems: shortcuts elaborated from fast-forward and time-reversed protocols
• Shortcuts to equilibrium with a levitated particle in the underdamped regime
• Non-Markovian Momentum Computing: Universal and Efficient
• Effects of a dissipative coupling to the momentum of a particle in a double well potential
• Shortcuts To Adiabaticity for Lévy processes in harmonic traps
• Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics

Cited by in corpus (6)

• Learning efficient erasure protocols for an underdamped memory
• Minimal work protocols for inertial particles in non-harmonic traps
• Optimal control of levitated nanoparticles through finite-stiffness confinement
• On the numerical integration of the Fokker-Planck equation driven by a mechanical force and the Bismut-Elworthy-Li formula
• Decomposition of metric tensor in thermodynamic geometry in terms of relaxation timescales
• Lower Bound of Entropy Production in an Underdamped Langevin System with Normal Distributions