Partial Open-Loop Feedback Control

• POLFC is a hybrid control architecture that combines extensive open-loop planning with selective local feedback to manage uncertainties and reduce computational load.
• The T-PFC formulation decouples the nominal trajectory from higher-order feedback gains, achieving near-optimal performance with significantly lower computational effort compared to NMPC.
• POLFC principles extend across diverse applications—from robotic planning and density control to quantum stabilization—demonstrating scalable, efficient control in complex stochastic systems.

Searching arXiv for POLFC and the cited papers to ground the article. Partial Open-Loop Feedback Control (POLFC) denotes a control architecture in which nominal planning is separated from local feedback regulation, especially under small noise, to avoid the curse of dimensionality in stochastic problems. In the cited literature, POLFC appears less as a single algorithm than as a recurring information structure: optimize or engineer as much of the control law as possible in open loop, then supplement it with reduced, local, scheduled, or minimally invasive feedback when open-loop structure alone is insufficient (Parunandi et al., 2019, Ticozzi et al., 2011, Bakshi et al., 2020).

1. Information structure and defining features

A basic distinction in the literature is between open-loop control, where the control protocol is set externally and independently of the system’s state, and feedback control, where the protocol is updated on the basis of measurements or state estimates. In time-dependent kinetic Monte Carlo (KMC), for example, an open-loop protocol is a parameter $\lambda(t)$ prescribed explicitly as a function of time, whereas a feedback protocol is updated at specified times using measured system states (Chittari et al., 2024). In density control of marked jump diffusions, open-loop control is described as a centralized controller broadcasting identical control signals to an ensemble of agents, in contrast to closed-loop policies that depend on local feedback information (Bakshi et al., 2020).

POLFC occupies an intermediate position. In the density-control setting, state-parameterized controls such as $u(t)=u_1(t)+x\,u_2(t)$ are presented as integrating some feedback using open-loop broadcast control, thereby trading off pure feedforward simplicity against the computational and infrastructural burden of full feedback (Bakshi et al., 2020). In stochastic quantum stabilization, a closely related hybrid strategy is stated explicitly: use open-loop Hamiltonian control as much as possible, taking advantage of engineered dissipation and measurement structure, and supplement minimally with feedback only when open-loop control cannot achieve global asymptotic stability (GAS) (Ticozzi et al., 2011).

This combination of precomputed structure and restricted feedback is the central identifying feature of POLFC. A plausible implication is that POLFC should be understood primarily through its information pattern—what is planned offline, what is regulated online, and what measurements are actually required—rather than through any single plant model or optimization routine.

2. Decoupling, perturbation feedback, and the T-PFC formulation

The most explicit formalization of POLFC in the supplied literature is "T-PFC: A Trajectory-Optimized Perturbation Feedback Control Approach" (Parunandi et al., 2019). T-PFC considers nonlinear, control-affine stochastic systems of the form

$$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$

with nominal trajectory

$$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$

Its key statement is a decoupling principle between the open loop plan and the closed loop feedback gains. In the backward expansion of the cost-to-go about the nominal trajectory, the first-order term

$G_t=L_t+G_{t+1}A_t$

determines open-loop sensitivity, while the second-order term

$$P_t=L_{tt}+A_t^\intercal P_{t+1}A_t-K_t^\intercal S_tK_t+G_{t+1}\otimes \tilde{R}_{t,xx}$$

determines feedback sensitivity. The crucial property is that the equation for $G_t$ does not involve $P_t$, so the optimal nominal sequence can be computed independently of the perturbation-feedback design. The resulting control law is

$$\mathbf{u}_t=\bar{\mathbf{u}}_t+K_t(\mathbf{x}_t-\bar{\mathbf{x}}_t),$$

with

$$K_t=-S_t^{-1}\left(B_t^\intercal P_{t+1}A_t+\left(G_{t+1}\otimes \tilde{R}_{t,xu}\right)^\intercal\right),$$

and the gains are obtained by backward recursion after the nominal has been computed (Parunandi et al., 2019).

This structure makes T-PFC a direct realization of POLFC: open-loop planning is performed once, or only rarely repeated, while feedback is used to regulate perturbations around the nominal. The paper states that the truncated linear feedback law is third-order near-optimal, with closed-loop cost difference from the true stochastic optimum of $u(t)=u_1(t)+x\,u_2(t)$0. It also states that T-PFC matches the cost performance of Nonlinear Model Predictive Control (NMPC) in several difficult robotic planning and control examples, while requiring much lesser computational effort, and the detailed summary reports up to $u(t)=u_1(t)+x\,u_2(t)$1 less computation than NMPC together with drastically fewer replans (Parunandi et al., 2019).

Theoretical and practical comparisons are central to the POLFC interpretation. Relative to NMPC, which solves a nonlinear optimal control problem at every step, T-PFC solves once over a planning horizon and replans only if the realized trajectory deviates too far from the nominal. Relative to ILQG/DDP and T-LQR, the claim is not merely computational economy: T-PFC is described as incorporating higher-order terms so as to obtain third-order near-optimality, whereas the comparison summary assigns lower-order optimality to standard LQR-type constructions (Parunandi et al., 2019). This suggests that, in the small-to-moderate noise regime, POLFC can be justified not only heuristically but by asymptotic optimality arguments.

3. Quantum manifestations: open-loop sufficiency and minimal-feedback stabilization

In quantum measurement and quantum state preparation, the supplied papers show two different ways in which POLFC-related ideas arise. The paper on rapid readout of a qubit register studies an open-loop protocol in which random permutations of the logical basis are applied during continuous measurement, without dependence on measurement outcomes (Combes et al., 2014). The stochastic master equation for $u(t)=u_1(t)+x\,u_2(t)$2 independently measured qubits is

$u(t)=u_1(t)+x\,u_2(t)$3

Using infidelity $u(t)=u_1(t)+x\,u_2(t)$4 and log-infidelity $u(t)=u_1(t)+x\,u_2(t)$5 as performance measures, the no-feedback asymptotic decay is stated as

$u(t)=u_1(t)+x\,u_2(t)$6

with mean hitting time

$u(t)=u_1(t)+x\,u_2(t)$7

Under random permutations, the speedup factor satisfies

$u(t)=u_1(t)+x\,u_2(t)$8

for large $u(t)=u_1(t)+x\,u_2(t)$9, and numerical simulations for small $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$0 yield approximately $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$1. The cited locally optimal feedback protocol has speedup approximately $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$2, so the open-loop method attains the same $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$3 scaling while being experimentally far less difficult (Combes et al., 2014).

Although this paper does not formulate the problem as POLFC, it is relevant because it demonstrates that open-loop structure can deliver performance of the same order as adaptive feedback. The commuting property is decisive: the permutation control commutes with the measured observable at all times, so the protocol does not alter which information is extracted, only the rate at which it is obtained. This suggests one pole of the POLFC spectrum: whenever plant or measurement structure makes open-loop actions effectively non-disruptive, partial or even zero feedback may be sufficient.

The quantum stabilization paper treats the complementary case, where feedback is necessary only when open-loop dissipation and measurement do not already create the needed coupling to the target subspace (Ticozzi et al., 2011). For a block decomposition $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$4, subspace invariance requires $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$5 for all $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$6 and

$$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$7

while GAS is possible in open loop if and only if there exists at least one $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$8 such that $$\mathbf{x}_{t+1}=f(\mathbf{x}_t)+g(\mathbf{x}_t)\,\mathbf{u}_t+\epsilon\sqrt{dT}\,\boldsymbol{\omega}_t,$$9. Corollary 4.1 states that if $$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$0 for all $$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$1, then GAS stabilization cannot be achieved with open-loop Hamiltonian control alone; feedback is then necessary (Ticozzi et al., 2011).

The resulting hybrid architecture is explicitly POLFC-like. Open-loop engineering does all it can, and filtering-based feedback is activated only to bridge otherwise uncoupled subspaces or eliminate unwanted invariant sets. The feedback law described for pure-state stabilization depends on the estimated overlap with the target state, and the summary emphasizes that only a small number of real-valued state parameters need to be tracked (Ticozzi et al., 2011). In this formulation, partial feedback is not merely a computational approximation; it is a consequence of the invariant-subspace structure of the controlled quantum dynamics.

4. Density control, scheduled feedback, and stochastic simulation

In large-population systems modeled by continuum densities, POLFC appears as a compromise between centralized broadcast control and full state feedback. The marked jump diffusion density-control paper formulates agent dynamics as a $$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$2-marked Markov jump diffusion and propagates the ensemble density through a forward Chapman–Kolmogorov partial integro-differential equation (PIDE) (Bakshi et al., 2020). The deterministic open-loop optimal control problem is

$$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$3

subject to the density evolution. Necessary optimality conditions are obtained from the infinite dimensional minimum principle (IDMP), with Hamiltonian

$$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$4

and the control is characterized by the stationarity condition $$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$5 (Bakshi et al., 2020).

The paper then makes the POLFC connection explicit. Open-loop control is described as computationally simple, scalable, and requiring minimal communication, but potentially suboptimal relative to full DPP/HJB feedback. POLFC is presented as the middle ground, using reduced-order or parameterized feedback while retaining the centralized broadcast structure. The example $$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$6 is given as a state-parameterized control that integrates some feedback using open-loop broadcast control (Bakshi et al., 2020). The associated Iteratively Backpropagated Costate algorithm is forward-sampling based and highly parallelizable, which places POLFC in a larger computational narrative: approximate the benefits of feedback without incurring the full PDE burden of closed-loop stochastic dynamic programming.

A closely related issue arises in the simulation of stochastic systems with time-dependent or feedback-dependent protocols. In the KMC paper, the term POLFC is not explicitly named, but the proposed feedback-controlled KMC method is said to interpolate between open-loop and fully closed-loop cases, with irregular measurement scheduling incorporating partial feedback-control aspects (Chittari et al., 2024). For open-loop control, the single-step trajectory probability is

$$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$7

For feedback control, Algorithm 2 introduces a milestone framework: between measurement times, the protocol is generated from past measurements, and beyond the next measurement time it is projected under the assumption that the system remains in its current state until the next jump (Chittari et al., 2024).

The significance for POLFC is methodological. When feedback is only scheduled at discrete measurement times, or when control updates are tied to milestone events rather than continuous observation, the information pattern becomes inherently partial. The KMC paper also shows that naive direct Gillespie implementations can generate an artificial Zeno effect, in which the system appears to freeze as measurement windows shrink, whereas the proposed method correctly integrates transition probabilities across piecewise protocols and avoids that artifact (Chittari et al., 2024). This suggests that POLFC is not only a controller-design issue but also a simulation issue: partial feedback must be represented with the correct trajectory probability structure.

5. From open-loop synthesis to feedback realization

Several supplied papers study how an open-loop solution can be realized, approximated, or learned as a closed-loop controller. In the sparse-feedback paper for discrete-time linear invariant systems, the starting point is a sparse optimal control problem minimizing $$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$8 subject to finite-horizon reachability constraints (Zhang et al., 2023). The feedback realization is constructed by a dynamic linear compensator,

$$\bar{\mathbf{x}}_{t+1}=f(\bar{\mathbf{x}}_t)+g(\bar{\mathbf{x}}_t)\,\bar{\mathbf{u}}_t.$$9

$G_t=L_t+G_{t+1}A_t$0

and the central result is that the implemented sparse feedback control is equivalent to the original sparse open-loop control solution under a specified basis. The resulting controller is an $G_t=L_t+G_{t+1}A_t$1-step deadbeat controller, and the compensator is extended to feedforward tracking through additional gains $G_t=L_t+G_{t+1}A_t$2 and $G_t=L_t+G_{t+1}A_t$3 (Zhang et al., 2023).

This formulation is closely aligned with POLFC, although the paper frames it as "from open-loop solution to closed-loop realization." The feedback law is not derived by solving a new online optimization problem; rather, a dynamic compensator is synthesized so that the closed-loop system reproduces the precomputed sparse open-loop sequence. A plausible implication is that one strand of POLFC consists in embedding open-loop optimality inside a realizable feedback architecture.

The learning-based paper on optimal feedback operators pushes this idea into function space (Kunisch et al., 2022). Instead of solving the Hamilton–Jacobi–Bellman equation directly, it poses a learning problem over $G_t=L_t+G_{t+1}A_t$4 such that the open-loop value function is the optimal solution. With polynomial ansatz

$G_t=L_t+G_{t+1}A_t$5

the finite-dimensional optimization is regularized by

$G_t=L_t+G_{t+1}A_t$6

and optimized באמצעות a proximal update with soft-thresholding. The paper proves convergence of polynomially parameterized policies to the true open-loop solution under increasing polynomial degree, increasing time horizon, and vanishing regularization, and it introduces a graph-theoretic algorithm for efficient evaluation of basis functions and derivatives (Kunisch et al., 2022).

Its relation to POLFC is stated cautiously in the summary: the approach resembles POLFC because the learning objective is constructed so that the open-loop value function is optimal, while the learned object is a feedback law. The difference is that optimization occurs directly over a feedback operator rather than by piecemeal interpolation of local open-loop solutions (Kunisch et al., 2022). In this sense, POLFC can be interpreted either as a runtime architecture or as an overview principle for converting open-loop optimality information into compact feedback representations.

The convection-cooling paper provides an analogous bridge in bilinear PDE control (Hu et al., 2021). Full open-loop optimal control requires forward solution of the state equation, backward solution of the adjoint equation, and iterative satisfaction of a nonlinear optimality condition. Closed-loop design is instead obtained by instantaneous control, with one-step optimization after time discretization. The resulting feedback law is

$G_t=L_t+G_{t+1}A_t$7

and the paper states that both discretize-then-optimize and optimize-then-discretize approaches yield the same feedback law under appropriate discretizations (Hu et al., 2021). The term POLFC is not used directly, but the summary states that choosing a short receding horizon produces a bridge between global open-loop optimization and fully local feedback. This suggests a horizon-based interpretation of POLFC, with instantaneous control at one extreme and full-horizon open-loop optimization at the other.

6. Mean-field games, performance trade-offs, and recurrent misconceptions

In linear-quadratic mean field Stackelberg games, the open-loop/feedback distinction becomes an information-structure issue in a hierarchical multi-agent setting. The leader announces a strategy first, and followers then optimize a social cost. Open-loop solutions are obtained by variational analysis with mean-field approximations, leading to controls expressed through mean-field forward-backward stochastic differential equations (FBSDEs) and coupled Riccati equations (Wang et al., 13 Apr 2025). Feedback solutions are derived through the matrix maximum principle, with decentralized gains of the form

$G_t=L_t+G_{t+1}A_t$8

Both open-loop and feedback solutions are shown to be asymptotic Stackelberg equilibria, and numerical simulations reported in the summary indicate that the feedback solution outperforms the open-loop solution for the followers’ average social cost under most parameter settings, whereas in some cases the open-loop strategy yields a lower cost for the leader (Wang et al., 13 Apr 2025).

This result is important because it blocks a common oversimplification: POLFC is not equivalent to “less feedback is always worse” or “more feedback is always better.” In the supplied literature, the preferred information structure depends on the task, noise level, coupling structure, measurement model, and objective. T-PFC explicitly questions why NMPC should be used in robotic control as opposed to perturbation-feedback approaches when near identical performance can be obtained with much lesser computational effort (Parunandi et al., 2019). By contrast, the density-control paper states that open-loop control may be suboptimal versus full feedback in response to stochastic state deviations, even though it has lower computational and infrastructural requirements (Bakshi et al., 2020). The quantum stabilization paper is sharper still: open-loop control is sufficient if and only if the dissipative or measurement structure couples the undesired subspace to the target; otherwise feedback is necessary (Ticozzi et al., 2011).

The most accurate general characterization is therefore conditional. POLFC is valuable when open-loop design captures most of the relevant structure and residual uncertainty can be handled by low-dimensional, local, intermittent, or scheduled feedback. It is less a doctrinal preference for open loop than a systematic search for the smallest feedback architecture compatible with the required stability, optimality order, robustness, or implementability.