Data-Driven Chance-Constrained Density Steering

• The paper introduces a framework that integrates data-enabled dynamics with chance constraints to steer state distributions accurately.
• It employs Gaussian and Gaussian-mixture models to represent multimodal uncertainties and optimize finite-horizon control performance.
• The approach balances probabilistic safety and control cost using tractable risk approximations and data-driven uncertainty estimation.

Data-driven chance-constrained density steering is the class of finite-horizon stochastic control problems in which a controller steers a state distribution while enforcing probabilistic safety constraints, with the dynamics or uncertainty representation obtained from data rather than assumed fully known a priori. In the current literature, the topic emerges from the intersection of density steering for linear systems and Gaussian mixtures, chance-constrained covariance steering, and data-driven predictive control or sample-based uncertainty modeling. The resulting formulations range from exact Gaussian or Gaussian-mixture endpoint steering to soft terminal distribution matching, and from explicitly model-based chance constraints to direct data-enabled or sample-driven safety layers (Nakashima et al., 8 Aug 2025, Balci et al., 2023, Okamoto et al., 2018, Haddad et al., 2020).

1. Scope and relation to covariance steering

In the cited work, density steering prescribes initial and terminal state distributions over a finite horizon and optimizes an expected control criterion. For discrete-time linear systems, one important special case is Gaussian endpoint steering, usually called covariance steering because a Gaussian law is determined by its mean and covariance. The finite-horizon chance-constrained covariance steering literature therefore forms a direct subcase of chance-constrained density steering, while Gaussian-mixture formulations extend the state law beyond a single mode (Okamoto et al., 2018).

A central distinction is that density steering targets the probability law itself, whereas chance constraints target the probability of remaining inside safe sets. This distinction matters because exact terminal density regulation does not, by itself, enforce pathwise safety. A separate line of work makes this explicit by solving density steering with trajectory chance constraints for non-Gaussian disturbances through characteristic functions, while another line uses predicted collision probabilities to construct a safe terminal PDF and then steers the system to that PDF without embedding a single explicit chance constraint inside the steering optimization (Sivaramakrishnan et al., 2021, Haddad et al., 2020).

The phrase “data-driven” is also heterogeneous in this literature. It can mean direct behavioral representations of unknown linear dynamics from persistently exciting state-input data, identification of finite-horizon multi-step predictors with quantified epistemic uncertainty, nonparametric or particle-based PDF estimation from online sensing, or distributionally robust chance constraints defined over Wasserstein balls around empirical distributions (Nakashima et al., 8 Aug 2025, Balim et al., 2024, Chen et al., 2022). This multiplicity of meanings is a structural feature of the field rather than a terminological anomaly.

2. Representative mathematical formulations

A model-based Gaussian-mixture density-steering formulation for discrete-time linear systems takes the form

$x_{k+1}=A_k x_k + B_k u_k,$

with initial and terminal state laws prescribed exactly as Gaussian mixture models: $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$ The optimization minimizes an expected quadratic cost over randomized feedback policies that map the current state to a probability distribution over controls (Balci et al., 2023).

A chance-constrained density-steering formulation with general disturbances uses

$$\mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k + D_k \mathbf{w}_k,$$

together with joint trajectory chance constraints for state and input on polytopic sets, and an approximate terminal density-steering term based on characteristic-function mismatch rather than exact endpoint equality. In its final computational form, the terminal distribution condition is softened into a penalty rather than enforced exactly (Sivaramakrishnan et al., 2021).

A direct data-driven chance-constrained density-steering formulation for an unknown LTI system starts from

$x_{t+1} = A x_t + B u_t,$

where $A$ and $B$ are constant but unknown and the uncertainty enters through a random initial condition. The initial state is modeled as

$$x_0 \sim \mathrm{GMM}\bigl(\{\alpha_i,\mu_0^i,\Sigma_0^i\}_{i=0}^{K-1}\bigr),$$

the terminal state is required to become Gaussian, and the state must satisfy the joint chance constraint

$$\mathbb{P}\biggl(\bigwedge_{j=1}^{L}\bigwedge_{t=1}^{N} x_t\in H_j\biggr)\ge 1-V, \qquad H_j := \{x\in\mathbb{R}^n\mid a_j^\top x \le b_j\}.$$

The objective combines expected control energy with a terminal Gaussian Gromov–Wasserstein penalty $\epsilon\,\mathrm{GGW}_2^2(\Sigma_d,\Sigma_N)$ (Nakashima et al., 8 Aug 2025).

A continuous-time alternative, motivated by automated driving, treats the joint state PDF as the controlled object and evolves it by the Liouville or regularized Fokker–Planck equation after differential-flatness-based transformation of the vehicle dynamics. There the endpoint PDFs are prescribed in flat coordinates, and safety enters through collision-probability-informed target selection rather than a direct chance constraint in the density-control layer (Haddad et al., 2020).

3. Density and uncertainty representations

Gaussian mixture models are the dominant finite-dimensional representation for multimodal density steering in the cited discrete-time linear literature. They are used because they “give good approximations for general smooth probability density functions,” and because suitably randomized policies can preserve the Gaussian-mixture class under controlled propagation. In the model-based setting, this structure reduces a finite-horizon Gaussian-mixture steering problem to a linear program after exact Gaussian-to-Gaussian covariance-steering subproblems are combined with a randomized policy over mixture components (Balci et al., 2023).

Characteristic functions provide a different representation, aimed at non-Gaussian uncertainty. For affine feedback and general disturbances, the achieved terminal density is not matched by moments alone; instead, the discrepancy is controlled through

$$D(K,v) = \left(\frac{1}{2\pi}\right)^n \int_{\mathbb{R}^n} \left|\varphi_{x_N}(t)-\varphi_{x_f}(t)\right|\,dt,$$

which upper-bounds the uniform pointwise mismatch between the achieved and desired terminal PDFs. The same characteristic-function machinery supports deterministic evaluation of scalar chance constraints through the Gil-Pelaez inversion theorem (Sivaramakrishnan et al., 2021).

A third representation is explicitly sample-based. In safe automated driving, transient PDFs are propagated by moving horizon nonparametric forecasts over weighted point clouds $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$0, obtained by propagating sampled initial conditions along closed-loop characteristics of the Liouville PDE rather than reconstructing PDFs from repeated Monte Carlo snapshots (Haddad et al., 2020).

A fourth representation is predictor-based rather than density-parametric. For uncertain linear systems with noisy output measurements, identified multi-step predictors yield

$$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$1

when the predictors are fixed, while the predictor parameters themselves are modeled as

$$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$2

This gives a data-driven finite-horizon distribution model with distinct aleatoric and epistemic components, even though the underlying control problem is framed as stochastic MPC rather than full density steering (Balim et al., 2024).

4. Chance constraints and tractable reformulations

For Gaussian state laws, chance constraints couple mean and covariance in a way that destroys the classical separation used in unconstrained covariance steering. In discrete-time linear stochastic systems with additive Gaussian disturbance, the deterministic reformulation of each probabilistic state constraint depends on both the trajectory mean and the state covariance, both of which depend on the same feedback matrix. This is the key reason that mean steering and covariance steering “cannot decouple” under chance constraints (Okamoto et al., 2018).

The tractable remedies differ across papers. One line conservatively decomposes joint chance constraints into individual facet-wise constraints using Boole-Bonferroni risk allocation and then exploits Gaussian quantiles, SOC structure, and an LMI-representable terminal covariance relaxation. The resulting problem is best described as a convex conic program with SDP and SOC structure rather than a purely unconstrained covariance-steering problem (Okamoto et al., 2018).

With partial observations, chance-constrained covariance control can be formulated under output feedback by reconstructing the filtered state with a Kalman filter, rewriting the chance constraints as difference-of-convex constraints, and solving the resulting DC program by successive convexification. That formulation explicitly addresses measurement noise and partial state availability, which are largely absent from the earlier fully observed steering literature (Pilipovsky et al., 2023).

Under multiplicative or parametric disturbances, a different tractable route uses Boole decomposition, Cantelli inequality, linear upper bounds on square-root terms, and Schur-complement LMIs. The resulting SDP yields a controller that is “valid (albeit potentially suboptimal)” for the original problem when the terminal covariance equality is relaxed to an upper bound. The paper is explicit that this pipeline is conservative for multiple reasons, including risk decomposition, Cantelli tightening, linearization, and covariance relaxation (Knaup et al., 2023).

In sample-driven settings, chance constraints can be approximated by deterministic nonlinear constraints built from biased kernel density estimators. The central sufficient condition is that the integrated kernel upper-bound the relevant indicator function pointwise, so that the deterministic KDE constraint does not violate the original probability bound when enough MCMC samples are available for law-of-large-numbers behavior (Keil et al., 2020).

Distributionally robust formulations over Wasserstein balls sharpen the cautionary picture. For DR chance constraints built from empirical distributions, the CVaR surrogate is characterized as a tight convex approximation in a specific geometric sense, but the CVaR, Bonferroni, and expected-violation approximations can all perform arbitrarily poorly in data-driven settings and are generally incomparable with each other (Chen et al., 2022).

5. Meanings of “data-driven” in the literature

One meaning of “data-driven” is online uncertainty estimation and nonparametric PDF forecasting. In automated driving, the ego vehicle estimates its own and neighboring vehicles’ state PDFs using tools such as a particle filter, propagates those PDFs over a short horizon through closed-loop Liouville characteristics, computes collision probabilities, and constructs a “safe” terminal PDF as a Wasserstein barycenter of a feasible traffic gap. The subsequent density-control layer then uses OMT and Schrödinger bridge machinery to steer the ego PDF to that target (Haddad et al., 2020).

A second meaning is sample-based approximation of the safety layer itself. The KDE method treats samples of the uncertainty vector $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$3 as the primitive object and replaces scalar chance constraints by smooth sample-average inequalities of the form

$$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$4

where $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$5 is an integrated biased kernel designed to preserve conservatism relative to the original bound (Keil et al., 2020).

A third meaning is sample-driven ambiguity modeling. In DR chance-constrained programs over Wasserstein balls, the uncertain law is represented by the empirical measure

$$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$6

and safety is required for all distributions within a Wasserstein radius $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$7 of $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$8. This line is not density steering per se, but it is directly relevant whenever chance-constrained steering is built on learned disturbance or trajectory samples rather than a known analytic law (Chen et al., 2022).

A fourth meaning is predictor identification. The multi-step predictive-control framework identifies finite-horizon predictors directly from open-loop input-output data using a surrogate innovation-form state-space model, then quantifies predictor uncertainty and derives an SOCP-based tightening that guarantees chance-constraint satisfaction despite parametric uncertainty. This produces a learned predictive distribution model even though the objective is quadratic tracking rather than explicit distribution matching (Balim et al., 2024).

A fifth meaning is direct data-enabled dynamics representation. In the direct data-driven density-steering line, the unknown linear dynamics are not first identified as $$x_0 \sim \mathbf{GMM}\!\left(\{p_i^0,\mu_i^0,\Sigma_i^0\}_{i=0}^{r-1}\right),\qquad x_N \sim \mathbf{GMM}\!\left(\{p_j^d,\mu_j^d,\Sigma_j^d\}_{j=0}^{t-1}\right).$$9; instead, persistently exciting state-input data are assembled into data matrices, and mean and covariance propagation constraints are written directly from those data. The theoretical development is explicit that this route is derived for the noiseless-data case (Nakashima et al., 8 Aug 2025).

6. Direct data-driven chance-constrained density steering and current research frontiers

The most explicit formulation of data-driven chance-constrained density steering in the cited corpus considers an unknown deterministic controllable LTI system, an initial Gaussian-mixture state distribution, half-space state chance constraints over the whole horizon, and a terminal Gaussian target evaluated through the Gaussian Gromov–Wasserstein metric. The control law is a mixture-dependent affine feedback,

$$\mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k + D_k \mathbf{w}_k,$$0

with

$$\mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k + D_k \mathbf{w}_k,$$1

Under this parameterization the state remains a GMM, terminal Gaussianity is enforced by requiring all terminal component means and covariances to coincide, and the unknown dynamics enter only through persistently exciting data matrices and a data-enabled mean/covariance recursion. The resulting optimization is written as a difference-of-convex program and solved by the DC algorithm, with each iteration requiring the solution of a convex SDP (Nakashima et al., 8 Aug 2025).

This direct data-driven construction is best understood as an overview of several earlier methodological strands. The Gaussian-mixture-preserving viewpoint echoes the model-based GMM steering literature, where randomized policies maintain mixture structure and make multimodal steering tractable (Balci et al., 2023). The explicit trajectory-level chance-constraint viewpoint aligns with the characteristic-function-based density-steering line, which treats state and input chance constraints under non-Gaussian disturbances but remains model-based and computationally nonlinear (Sivaramakrishnan et al., 2021). The separation between probabilistic prediction and density control also remains visible in applications such as automated driving, where a safe target PDF is constructed from predicted collision probabilities before density steering is applied (Haddad et al., 2020).

Several limitations recur across the literature. Exact or approximate terminal density steering is not equivalent to pathwise probabilistic safety; probabilistic safety must be imposed or synthesized separately. Data-driven safety surrogates can be conservative, and Wasserstein-ball approximations can be mutually incomparable rather than hierarchically ordered (Chen et al., 2022). Direct behavioral steering from data currently relies on noiseless-data theory, while learned-predictor approaches provide tractable probabilistic guarantees at the price of conservatism. In the multi-step predictor framework, for example, the reported output chance-constraint violation under the proposed data-driven method is $$\mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k + D_k \mathbf{w}_k,$$2, compared with $$\mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k + D_k \mathbf{w}_k,$$3 for the exact known-model stochastic MPC at $$\mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k + D_k \mathbf{w}_k,$$4, indicating safe but conservative tightening (Balim et al., 2024).

A plausible implication is that the field is converging on hybrid architectures rather than a single dominant formulation. The recurring ingredients are data-enabled finite-horizon uncertainty propagation, density representations richer than a single Gaussian, and tractable but conservative risk surrogates. What remains unsettled is not whether these ingredients can be combined—they already have been—but which combination best balances exact distribution shaping, probabilistic safety, data dependence, and computational tractability across different problem classes.