CoSteer: Covariance Steering in Control

• CoSteer is a control method that designs feedback policies to explicitly steer both the mean and covariance of stochastic state trajectories subject to uncertainty.
• It employs affine feedback structures and convexification techniques to handle terminal distribution requirements and chance constraints in linear and nonlinear dynamics.
• Applications span robotics, aerospace, and multi-agent systems, while a separate use in language modeling underlines the term's diverse methodological innovations.

CoSteer is used most commonly in contemporary control literature as shorthand for covariance steering or constrained covariance steering: the synthesis of feedback policies that explicitly steer both the mean and the covariance of a stochastic state trajectory, typically under Gaussian assumptions, terminal distribution requirements, chance constraints, and finite-horizon performance criteria. In a separate and unrelated line of work, the same name was later used for a decoding-time personalization framework for LLMs. The dominant technical meaning, however, is the control-theoretic one, where CoSteer sits at the intersection of stochastic optimal control, distribution steering, chance-constrained optimization, and model predictive control (Ridderhof et al., 2021, Lv et al., 7 Jul 2025).

1. Terminology and scope

In the material represented here, the term has two distinct uses.

Usage	Meaning	Representative paper
CoSteer in control	Covariance steering or constrained covariance steering: explicit control of state-distribution mean and covariance under uncertainty	(Ridderhof et al., 2021)
"CoSteer" in language modeling	Collaborative decoding-time personalization via local delta steering between a cloud LLM and an on-device model	(Lv et al., 7 Jul 2025)

Within stochastic control, covariance steering is described as designing feedback controllers that explicitly shape both the mean and the covariance of a stochastic state trajectory, typically Gaussian, subject to constraints and performance criteria. That usage encompasses exact and relaxed terminal distribution steering, chance-constrained path planning, trajectory-distribution control inside sampling-based MPC, distributed multi-agent formulations, hybrid systems with jumps, and robustness to multiplicative, spatially correlated, or parametric uncertainty (Ridderhof et al., 2021).

A recurring distinction in this literature is between mean steering and covariance steering. Feedforward variables determine the nominal trajectory, while feedback gains determine the dispersion of the closed-loop state. This separation is central to most CoSteer formulations, even when the implementation differs—state-history feedback, memoryless affine feedback, disturbance-feedback parameterizations, or square-root covariance factorizations all preserve that basic division of roles.

2. Canonical formulation in stochastic control

A canonical discrete-time formulation uses linear or linearized stochastic dynamics together with an affine feedback law. One common policy is memoryless affine state feedback,

$u_k = v_k + K_k(x_k - E[x_k]),$

under which the mean and covariance evolve as

$$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$

Another formulation, motivated by temporally correlated disturbances induced by spatial uncertainty, uses state-history feedback,

$$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$

with $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$ (Ridderhof et al., 2021, Kumagai et al., 29 Jan 2026).

Terminal distribution requirements are the defining constraint class. In one representative form, CoSteer enforces

$$\mathbb{E}(x_f)=\bar{x}_f,\qquad \mathrm{Cov}(x_f)\preceq P_f,$$

while other formulations impose exact covariance assignment or a Wasserstein-distance penalty to a target Gaussian. Exact equality and positive-semidefinite upper bounds coexist in the literature, with the latter often used to recover convexity or improve feasibility (Ridderhof et al., 2021, Balci et al., 2022).

Chance constraints are typically imposed on affine functions of Gaussian state or control variables. Because linear functionals of Gaussian random vectors remain Gaussian, one-sided constraints admit deterministic reformulations of the form

$\mu + \Phi^{-1}(1-p)\sigma \le \alpha,$

which become second-order cone constraints. This is the standard route for probabilistic bounds on state, control, altitude, heating, actuator saturation, or obstacle-avoidance half-spaces (Ridderhof et al., 2021).

When free space is non-convex, CoSteer formulations commonly replace direct obstacle avoidance by unions of convex regions or local supporting half-spaces. In stochastic vehicle path planning, the safe region is decomposed into convex regions, Boolean variables select which region is active, and the resulting chance-constrained problem becomes a mixed-integer second-order cone program. The control law

$u_k = v_k + K_k y_k$

then allows the planner to optimize the nominal path and the uncertainty tube simultaneously rather than treating covariance as an open-loop byproduct (Okamoto et al., 2018).

3. Convexification and numerical solution methods

A major thread in CoSteer research is the search for formulations that preserve convexity or recover it through suitable transformations. For discrete-time linear systems with mixed multiplicative and additive noise, the relaxed covariance-steering problem—terminal covariance enforced as $\mathrm{Cov}(x_N)\preceq \Sigma_d$—can be recast as an equivalent convex semidefinite program after introducing variables such as

$$L_k := K_k\Sigma_k,\qquad \mathbf{M}_k := K_k\Sigma_kK_k^\top,$$

together with lifted mean and input outer-product variables. The paper distinguishes this relaxed problem from exact covariance assignment and proposes a two-step procedure for the exact case; when the SDP relaxation is not tight, a randomized affine policy may be needed to reproduce the optimal input covariance (Balci et al., 2022).

A second line of work replaces older block-matrix constructions by full-covariance decision variables and lossless convexification. In robust cislunar low-thrust trajectory optimization, the covariance recursion

$$\hat{P}_{k+1} = A_k\hat{P}_k A_k^\top + B_kU_kA_k^\top + A_kU_k^\top B_k^\top + B_kY_kB_k^\top + \hat{Q}_{k+1}$$

becomes affine after the change of variables

$$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$0

with consistency enforced through the LMI

$$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$1

This yields a full-covariance, lossless convexification compatible with $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$2 surrogates and chance-constrained SCP (Kumagai et al., 4 Feb 2025).

A more recent approach factorizes the covariance itself. Square Root-Factorized Covariance Steering writes the propagation equations of the Cholesky factor of the state covariance matrix by using the QR decomposition. The stated benefits are computational scalability and numerical reliability relative both to large block-matrix formulations and to methods that optimize the covariance matrix directly. The trade-off is non-convex square-root propagation, handled by sequential convex programming. The paper proves global optimality without chance constraints and states that, when chance constraints are present, the square-root formulation shares the same local minima as the existing optimal CS formulation (Kumagai et al., 29 Jan 2026).

Sequential convex programming is therefore not an auxiliary detail but one of the main computational idioms of modern CoSteer. It appears when dynamics are nonlinear, when chance constraints contain non-convex surrogates, when covariance is propagated through square-root factors, and when quantile objectives such as $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$3 are optimized rather than simple quadratic expectations.

4. Generalizations of uncertainty and dynamics

One major generalization replaces temporally white disturbance models by spatially dependent uncertainty. In chance-constrained covariance steering in a Gaussian random field, the disturbance is sampled through a state-dependent map $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$4, so motion through space induces a temporally correlated Gaussian process along the nominal trajectory. After linearization and discretization, the disturbance vectors $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$5 are jointly Gaussian with mean and covariance obtained from the field mean $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$6 and kernel $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$7. This extends CoSteer beyond i.i.d. or Brownian-noise models and motivates state-history feedback rather than purely Markov feedback (Ridderhof et al., 2021).

A second generalization addresses unknown constant parameters that enter multiplicatively in the dynamics. One formulation models these parameters as random variables with known moments, derives a moment-based representation involving joint moments such as $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$8, and solves the resulting non-convex problem with sequential convex programming. A later development adds online parameter estimation: recursive least squares updates $$\mu_{k+1} = A_k\mu_k + B_k v_k,\qquad P_{k+1} = (A_k + B_kK_k)P_k(A_k + B_kK_k)^\top + G_kG_k^\top.$$9, and the control law is made affine in both the current state and the current parameter estimate,

$$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$0

That formulation is explicitly dual: the control both steers the state distribution and shapes future parameter estimates (Knaup et al., 2023, Knaup et al., 2024).

Hybrid systems introduce another kind of departure from standard CS assumptions. In Hybrid Covariance Steering, the jump map at a mode transition is linearized via the Saltation Matrix,

$$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$1

which implies covariance propagation

$$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$2

When $$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$3 is square and invertible, the paper derives a closed-form solution through hybrid Riccati equations and a hybrid Hamiltonian flow. When $$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$4 is singular or rectangular, possibly with state-dimension changes, the problem is reformulated as a convex optimization over path distributions by exploiting Schrödinger bridge duality and imposing covariance propagation at jumps as equality constraints (Yu et al., 2024).

These extensions suggest that CoSteer is best understood not as a single algorithm but as a design pattern: formulate uncertainty evolution explicitly at the distribution level, choose a control parameterization that separates mean and covariance effects as much as possible, and recover tractability by convexification, factorization, or path-distribution reformulation.

5. Integration with MPC and decentralized optimization

CoSteer has been repeatedly fused with model predictive control and sampling-based control. In Constrained Covariance Steering Based Tube-MPPI, MPPI provides a performance-oriented nominal trajectory in a non-convex environment, while constrained covariance steering computes a disturbance-feedback controller that tracks that reference inside a probabilistic safe tube defined by obstacle-separating half-spaces. The resulting CCSMPPI algorithm uses MPPI for nominal planning, a half-space generator for local convexification of obstacles, and an SOCP for the covariance-steering subproblem. In obstacle avoidance and circular-track experiments, this combination substantially lowered empirical failure probabilities relative to standard MPPI and Tube-MPPI (Balci et al., 2021).

A related synthesis appears in Covariance-Controlled MPPI, where covariance steering is used to control the dispersion of MPPI rollouts at the end of the prediction horizon. The sampled control takes the form

$$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$5

with $$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$6 obtained from a covariance-steering problem enforcing a terminal covariance bound. This directly shapes the trajectory distribution used by MPPI. In aggressive driving experiments, the reported average lap time was $$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$7 s for CC-MPPI versus $$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$8 s for baseline MPPI, with success rates $$u_k = \sum_{\ell=0}^k K_{k,\ell}\,\tilde{x}_\ell + v_k,$$9 and $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$0, respectively (Yin et al., 2021).

Distributed Model Predictive Covariance Steering extends the framework to multi-agent systems. There the objective combines control effort with Wasserstein distance between each agent’s state distribution and a target Gaussian, while safety is approximated through probabilistic separation constraints between means and confidence ellipsoids. A disturbance-feedback policy parameterization yields closed-form mean and covariance expressions, and inter-agent coupling is handled by local copies and consensus constraints in ADMM. The resulting decentralized MPC scheme was demonstrated on tasks with up to hundreds of robots, and the paper states that hardware results on a multi-robot platform verify applicability (Saravanos et al., 2022).

Across these MPC integrations, a common architectural pattern emerges: nominal planning is handled by a receding-horizon or sampling-based outer loop, while covariance shaping is delegated to a structured inner problem whose solution determines the uncertainty tube, rollout dispersion, or inter-agent safety margin.

6. Aerospace and robotic applications

Robotic path planning provided one of the clearest early demonstrations of why covariance steering matters. In stochastic vehicle path planning with non-convex obstacles, the planner can actively shrink covariance to pass through a narrower but shorter slit, or choose a wider detour when reducing covariance is too costly. Because uncertainty is controlled directly rather than left to a lower-level stabilizer, the nominal path can be less conservative while still satisfying chance constraints (Okamoto et al., 2018).

Atmospheric aerocapture is a canonical aerospace application. In the Gaussian-random-field formulation, atmospheric density uncertainty is modeled as a spatial Gaussian random field over altitude, and the control law is optimized to minimize approximately the 99th percentile of terminal $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$1. Starting from a zero-control nominal and using three SCP iterations, Monte Carlo evaluation with $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$2 runs per policy reported a 99th-percentile $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$3 of approximately $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$4 m/s for the open-loop policy and approximately $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$5 m/s for the final closed-loop covariance-steering policy (Ridderhof et al., 2021).

Cislunar low-thrust mission design extends the same logic to highly nonlinear CR3BP dynamics, navigation errors, execution errors, and lunar flybys. There the controller jointly optimizes a nominal trajectory and a trajectory-correction policy while minimizing an approximate $\tilde{x}_\ell = x_\ell - \bar{x}_\ell$6. The reported DRO–DRO transfer converged in 6 iterations in about 10 seconds, while the 3D NRHO–Halo transfer converged in approximately 60 iterations in approximately 10 minutes with 200 nodes. The robust mean trajectory deviated from the deterministic reference near lunar flyby toward a less sensitive region, and the local Lyapunov exponent peaks were reduced relative to the deterministic design (Kumagai et al., 4 Feb 2025).

Hybrid locomotion and contact-rich motion supply a different application class. In the bouncing-ball and linearized SLIP examples of Hybrid Covariance Steering, CoSteer corrects the terminal covariance that hybrid iLQR alone does not control, including cases where the jump changes state dimension and induces singular post-impact covariance (Yu et al., 2024).

Taken together, these applications show that CoSteer is particularly valuable when the dominant engineering question is not merely how to hit a target mean trajectory, but how to arrive there with a prescribed uncertainty geometry.

7. Reuse of the name outside covariance control

In 2025 the title “CoSteer” was reused for an unrelated language-model framework: collaborative decoding-time personalization via local delta steering. In that setting, a cloud LLM produces base logits, an on-device small model computes the logits difference between personal-context-aware and personal-context-agnostic decoding, and a token-level online-learning update steers the cloud model locally without transmitting either the personal data or the steering vector. The update is formulated with FTRL-style regularization, and a one-step approximation,

$\tilde{x}_\ell = x_\ell - \bar{x}_\ell$7

is presented as LightCoSteer (Lv et al., 7 Jul 2025).

That NLP usage is terminologically separate from covariance steering. Its “steering” concerns logits in decoding-time personalization rather than state distributions in stochastic control. The overlap is therefore nominal rather than methodological. In the control literature, CoSteer remains the established shorthand for covariance steering: explicit control of mean and covariance, usually under Gaussian uncertainty, often under chance constraints, and increasingly across nonlinear, distributed, hybrid, and learning-augmented settings.