Equivalence classes of finite-time transitions in optimal control and non-equilibrium relaxation

Technical Summary: Equivalence Classes of Finite-Time Transitions in Optimal Control and Non-Equilibrium Relaxation

Problem Statement
The paper addresses the optimal control of stochastic systems in structured environments, specifically focusing on the trade-off between the speed of a process and its energetic cost. While optimal control theory provides a framework to minimize cost functionals over finite time horizons, the presence of spatially varying cost landscapes (e.g., obstacles or energy penalties) introduces complex, non-trivial behaviors. The authors investigate whether these control problems exhibit sharp, finite-time transitions in the optimal strategy as the protocol duration varies, and how these transitions relate to non-equilibrium relaxation phenomena, specifically finite-time dynamical phase transitions (FTDPTs).

Methodology
The authors develop a theoretical framework starting from a nonlinear overdamped Langevin equation for a particle in an inhomogeneous environment.

1. Linearization: They assume the existence of a "zero-control fixed point" (a stable equilibrium in the absence of active control) and expand the dynamics and a general quadratic cost functional around this point. This yields a linearized Langevin equation with additive noise and a quadratic cost functional involving the mean trajectory $u(t)$ and control $\lambda(t)$.
2. Hamiltonian Formulation: Applying the Pontryagin minimum principle, the optimization problem is mapped to a Hamiltonian system. The control Hamiltonian $H$ is a $2\times2$ matrix acting on the phase space of the mean coordinate and its conjugate momentum.
3. Canonical Classification: Through symplectic (canonical) transformations, the authors demonstrate that all such linearized control problems fall into three distinct equivalence classes, determined solely by the sign of the determinant of the control Hamiltonian, $\det H$ (or equivalently, the sign of a parameter $\zeta$ related to the cost matrix):
   • Parabolic Class ($\det H = 0$): Corresponds to free motion (e.g., mean work minimization in a harmonic trap).
   • Hyperbolic Class ($\det H < 0$): Corresponds to motion in an inverted parabola (e.g., quadratic control cost).
   • Elliptic Class ($\det H > 0$): Corresponds to harmonic oscillation (e.g., cost functions rewarding displacement from the origin).
4. Analytical Solution: For each class, the authors derive closed-form expressions for the optimal protocols, the cost functions, and the critical times at which transitions occur.
5. Mapping to Relaxation: A duality is established between the optimal control cost and the large-deviation rate function governing non-equilibrium relaxation after a potential quench. This mapping involves time reversal, boundary condition exchange, and a harmonic shift.
6. Experimental Validation: The theory is tested using optically trapped colloidal particles. Three experiments are conducted: (i) direct measurement of the control transition in the parabolic class; (ii) measurement of the associated FTDPT in free diffusion (parabolic); and (iii) measurement of the FTDPT in harmonic relaxation (hyperbolic). The relaxation experiments utilize an importance-sampling technique (reweighting unconditioned trajectories) to access rare events without exponential sampling costs.

Key Contributions and Results

• Universal Classification: The paper establishes that optimal control problems with quadratic costs and linearized dynamics are universally categorized into three classes (parabolic, hyperbolic, elliptic). This classification dictates the functional form of the optimal protocol (linear, hyperbolic, or trigonometric) and the nature of the control transition.
• Finite-Time Control Transitions: The authors identify a sharp transition in the optimal control strategy at a critical time $t_c$. Below $t_c$, the optimal strategy keeps the particle at the zero-control position (or follows a trivial path). Above $t_c$, the optimal strategy spontaneously breaks symmetry (in symmetric potentials) or shifts to avoid high-cost regions, resulting in a non-zero optimal final position.
• Landau Analogy: The transition is shown to be analogous to a continuous phase transition in mean-field theory. The optimal final position acts as an order parameter, growing as $(t_f - t_c)^{1/2}$ near the critical time, yielding a mean-field critical exponent $\beta_{MF} = 1/2$. The optimal cost function develops a "kink" at the critical initial position, analogous to a free energy singularity.
• Control-Relaxation Duality: A rigorous mapping is derived linking the optimal control cost to the large-deviation rate function of non-equilibrium relaxation.
  • This mapping holds for the parabolic (free diffusion) and hyperbolic (harmonic relaxation) classes.
  • The elliptic class has no simple relaxation counterpart because it requires an imaginary trap stiffness in the relaxation picture and exhibits instability (unbounded negative cost) beyond a finite time.
• Experimental Verification:
  • Parabolic Control: Experiments confirmed the transition in mean stochastic work, observing the kink in the cost vs. initial position and the square-root growth of the order parameter.
  • Relaxation FTDPTs: Using reweighting, the authors reconstructed rate functions for free and harmonic relaxation. They observed the emergence of kinks in the rate functions and the reorganization of the order-parameter density.
  • Critical Fluctuations: A key distinction was found: while the control transition is deterministic and sharp at any noise level, the FTDPT in relaxation exhibits genuine critical fluctuations. The susceptibility of the order parameter in relaxation diverges as the noise vanishes, following mean-field scaling laws ($\chi \propto |t_f - t_c|^{-1}$), a feature absent in the control experiment.

Significance
The paper claims to provide a complete theoretical classification of finite-time transitions in optimal control for stochastic systems in structured environments. Its primary significance lies in:

1. Unifying Framework: It connects diverse control problems (mean work, quadratic control) under a single Hamiltonian classification scheme, revealing that their qualitative behaviors are dictated by the geometry of the control Hamiltonian.
2. Bridging Control and Rare Events: It establishes that optimal control experiments can access the same physical information as finite-time dynamical phase transitions (FTDPTs) in non-equilibrium relaxation. Crucially, the control approach bypasses the exponential sampling cost typically required to observe rare relaxation events directly.
3. Experimental Accessibility: By demonstrating that FTDPTs can be probed via controlled trajectories (which are easy to sample) or via reweighting techniques, the work offers a practical route to studying rare-event physics and dynamical phase transitions in mesoscopic systems without the prohibitive computational or experimental costs of direct rare-event sampling.
4. Distinction of Phenomena: The work clarifies that while control transitions and FTDPTs are mathematically dual, they are physically distinct: the former is a deterministic optimization result, while the latter involves stochastic critical fluctuations.

The authors conclude that finite-time control transitions are a generic feature of optimal control in structured mesoscopic environments and that the proposed classification and mapping provide a robust framework for understanding and designing such processes.