Control Density Functions (CDFs)

• Control Density Functions (CDFs) are scalar fields that represent state occupancy, dissipation, and spatial importance for synthesizing control laws.
• They are applied in nonlinear stabilization, safety-critical navigation, and multi-agent coverage using methods like MPC, SOS programming, and CVT-based deployment.
• Recent research shows that CDF frameworks can improve safety margins and computational efficiency by shifting from trajectory-centric to density-based formulations.

Searching arXiv for recent and foundational papers on control density functions and related density-based control formulations. Control Density Functions (CDFs) denote several technically distinct but conceptually related uses of density-like objects in control, robotics, and learning. In control theory proper, the term refers most directly to control dissipation functions for discrete-time nonlinear stabilization (Lazar, 2021), and, in a separate density-theoretic lineage, to density functions used as dual certificates for safety, navigation, and reach-avoid synthesis (Zheng et al., 2023, Bozdag et al., 21 May 2025, Narayanan et al., 16 Sep 2025). In multi-robot and coverage control, density functions encode spatial importance and induce target distributions for agent deployment [(Lee et al., 2014); (Song et al., 22 Oct 2025)]. Across these usages, a recurring theme is a shift away from purely trajectory-centric formulations toward objects that encode occupancy, dissipation, or spatial priority in a way that supports controller synthesis, analysis, or distributed coordination.

1. Terminological scope and major lineages

The phrase “Control Density Functions” is not attached to a single universal formalism. The supplied literature exhibits at least three distinct usages.

First, control dissipation functions are introduced for discrete-time nonlinear systems subject to state/input constraints. In this setting, a CDF is a storage function $V$ for which there exists a feedback control law satisfying a dissipation inequality with an associated supply function $s$ on a constrained controlled invariant set (Lazar, 2021).

Second, a density-function lineage treats densities as dual objects to Lyapunov or barrier formulations for safety and navigation. “Safe Navigation using Density Functions” constructs density functions analytically for almost everywhere navigation with safety constraints, using the dual formulation of the navigation problem (Zheng et al., 2023). “Safe Control for Pursuit-Evasion with Density Functions” extends this perspective to reach-avoid pursuit-evasion differential games with dynamic unsafe sets, treating the pursuer as an adversarial disturbance and synthesizing safe control through density functions and convex SOS programs (Bozdag et al., 21 May 2025). “Safety Critical Model Predictive Control Using Discrete-Time Control Density Functions” embeds such density constraints in MPC and emphasizes an occupancy interpretation of the associated measure (Narayanan et al., 16 Sep 2025).

Third, in coverage control and multi-robot coordination, density functions specify where agents should be located. “Multi-Robot Control Using Time-Varying Density Functions” treats density functions as rough references for robot location over time (Lee et al., 2014), while “Multi-UAV Flood Monitoring via CVT with Gaussian Mixture of Density Functions for Coverage Control” uses a control density function $\phi(\boldsymbol{q}, t)$ to encode spatial importance in the workspace and drive CVT-based deployment (Song et al., 22 Oct 2025).

This suggests that “CDF” is best understood as a family resemblance term rather than a single definition. A plausible implication is that the common substrate is the use of a scalar field over state or workspace to encode control-relevant structure—stability budget, safety occupancy, or coverage priority.

2. Dissipativity-based CDFs for stabilization

In “Stabilization of discrete-time nonlinear systems based on control dissipation functions,” the system is

$x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$

and the core inequality is

$$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$

Here $V : \mathbb{R}^n \to \mathbb{R}$ is the storage function and $s : \mathbb{R}^n \to \mathbb{R}$ is the supply function (Lazar, 2021).

A storage function $V$ is a control dissipation function (CDF) in a set $X$ if there exists a feedback law $u(k)=\kappa(x(k))$, with $s$0, such that $s$1 is constrained controlled invariant under $s$2 and the dissipation inequality holds for all trajectories starting in $s$3 (Lazar, 2021). When $s$4 is positive definite and $s$5 is negative definite, the construction coincides with a classical control Lyapunov function, but the CDF framework allows more general non-monotone behavior through the supply term.

The central relaxation is the cyclically negative supply function condition,

$s$6

for some $s$7 and class-$s$8 function $s$9 (Lazar, 2021). Thus, $\phi(\boldsymbol{q}, t)$0 need not decrease at every step; instead, decrease is enforced over cycles. The paper proves that positive definite storage functions and cyclically negative supply functions yield asymptotic Lyapunov stability (Lazar, 2021).

For controller synthesis, the paper constructs stabilizing receding horizon controllers by minimizing CDFs subject to a cyclically negative supply condition. A Massera-type construction uses

$\phi(\boldsymbol{q}, t)$1

with positive definite kernel $\phi(\boldsymbol{q}, t)$2, and the online optimization minimizes this finite-horizon storage subject to dynamics and constraints (Lazar, 2021). The resulting formulation unifies dissipativity, finite-step CLFs, and MPC-style stabilization.

This line differs markedly from the navigation and safety papers: the density notion is not spatial occupancy but dissipativity. Nonetheless, both lines replace strict pointwise decrease conditions with more global certificate structures.

3. Density functions as dual certificates for navigation and safety

A second major usage employs density functions as dual objects for navigation and safe control. “Safe Navigation using Density Functions” explicitly formulates navigation in the dual space of densities rather than through potential fields or navigation functions (Zheng et al., 2023). Obstacles $\phi(\boldsymbol{q}, t)$3 are described by smooth functions $\phi(\boldsymbol{q}, t)$4, transition regions by $\phi(\boldsymbol{q}, t)$5, and the density is constructed as

$\phi(\boldsymbol{q}, t)$6

where $\phi(\boldsymbol{q}, t)$7 is a positive smooth distance-like function to the target, $\phi(\boldsymbol{q}, t)$8 is a smooth inverse bump function, and parameters $\phi(\boldsymbol{q}, t)$9 and $x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$0 tune convergence sharpness and positivity outside obstacles (Zheng et al., 2023). The induced safe feedback law is

$x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$1

The theoretical argument relies on divergence conditions such as

$x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$2

together with an occupancy-measure interpretation: enforcing $x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$3 to be zero inside obstacles ensures that no trajectory with nonzero measure occupies those sets (Zheng et al., 2023). The paper proves almost everywhere navigation: for all initial conditions except a set of measure zero, trajectories reach the target while avoiding the unsafe sets (Zheng et al., 2023).

“Robust Data-Driven Safe Control using Density Functions” uses the same dual perspective for unknown polynomial dynamics learned from noisy data. Instead of barrier conditions, which become non-convex under joint control synthesis, the paper seeks a density function $x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$4 satisfying

$x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$5

and extends this to robust controlled systems by introducing $x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$6, yielding a rational controller $x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$7 (Zheng et al., 2023). This formulation turns safety synthesis into a tractable convex optimization via SOS techniques and the theorem of alternatives (Zheng et al., 2023).

The same occupancy-based interpretation appears in “Safety Critical Model Predictive Control Using Discrete-Time Control Density Functions,” which states that the associated measure signifies the occupancy of system trajectories (Narayanan et al., 16 Sep 2025). This continuity across papers indicates a coherent dual-control perspective: density functions describe how trajectories distribute over state space, and safety is enforced by shaping that distribution rather than only constraining state evolution pointwise.

4. Reach-avoid pursuit-evasion and convex SOS synthesis

“Safe Control for Pursuit-Evasion with Density Functions” extends density-based safe control to a reach-avoid pursuit-evasion differential game with a dynamic unsafe set that depends on both evader and pursuer states (Bozdag et al., 21 May 2025). The joint state is

$x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$8

with dynamics

$x(k+1) = f(x(k), u(k)), \qquad k \in \mathbb{N}$9

where $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$0 is the evader control and $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$1 is the pursuer control treated as a disturbance (Bozdag et al., 21 May 2025).

The foundational density-function sufficient conditions for weak eventuality and safety are stated as

$$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$2

The paper then introduces an adversarial extension with variables $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$3, $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$4, shaping function $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$5, and scalar $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$6, together with the synthesized control law

$$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$7

If the inequalities are satisfied for all $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$8, then evasion and reachability hold robustly for almost all initial states (Bozdag et al., 21 May 2025).

A key technical contribution is the conversion of these PDE- and inequality-based conditions into a convex sum-of-squares program under polynomial dynamics, polynomial certificates, and semi-algebraic set descriptions. Positivity over semi-algebraic sets is relaxed using Putinar’s Positivstellensatz, and robustness with respect to all admissible pursuer controls is enforced using Farkas’ lemma (Bozdag et al., 21 May 2025). The paper states that the joint search for $$V(x(k+1)) - V(x(k)) \leq s(x(k)), \qquad \forall k \in \mathbb{N}.$$9, $V : \mathbb{R}^n \to \mathbb{R}$0, and SOS multipliers becomes a convex semidefinite feasibility problem solvable by standard SDP tools such as YALMIP+MOSEK (Bozdag et al., 21 May 2025).

The stated comparison with Hamilton-Jacobi-Isaacs methods is precise: HJI approaches require a value function PDE with min-max structure, grid-based discretization, and become computationally intractable for more than 4–5 dimensions, whereas the density-function-plus-SOS formulation is convex in the transformed variables, directly incorporates robustness, avoids viscosity-solution computation, and handles semi-algebraic constraints naturally (Bozdag et al., 21 May 2025).

This establishes one of the clearest meanings of “control density functions” in contemporary nonlinear control: density certificates that enable safe controller synthesis in problems where HJI formulations are classical but expensive.

5. Discrete-time MPC-CDF and safety-critical navigation

“Safety Critical Model Predictive Control Using Discrete-Time Control Density Functions” develops an MPC-CDF framework in which CDFs are used inside a discrete-time optimization problem for safety-critical control (Narayanan et al., 16 Sep 2025). The density construction follows the obstacle/product-over-target-distance pattern: $V : \mathbb{R}^n \to \mathbb{R}$1 where each obstacle is represented by an unsafe set $V : \mathbb{R}^n \to \mathbb{R}$2, a sensing region $V : \mathbb{R}^n \to \mathbb{R}$3, and an inverse bump function $V : \mathbb{R}^n \to \mathbb{R}$4 that is $V : \mathbb{R}^n \to \mathbb{R}$5 in the obstacle, smoothly transitions inside the sensing region, and is $V : \mathbb{R}^n \to \mathbb{R}$6 outside it (Narayanan et al., 16 Sep 2025).

The paper interprets the measure associated with $V : \mathbb{R}^n \to \mathbb{R}$7 as the total time or probability trajectories spend in a set. Obstacles are encoded by $V : \mathbb{R}^n \to \mathbb{R}$8, while the target corresponds to maximal density (Narayanan et al., 16 Sep 2025). For the discrete-time system

$V : \mathbb{R}^n \to \mathbb{R}$9

the paper invokes the Perron-Frobenius operator and presents the discrete-time CDF condition

$s : \mathbb{R}^n \to \mathbb{R}$0

then introduces an Euler-discretized local constraint

$s : \mathbb{R}^n \to \mathbb{R}$1

This constraint is then embedded into the MPC problem together with the dynamics, state/input constraints, and terminal set constraint (Narayanan et al., 16 Sep 2025).

The resulting optimizer minimizes a standard stage-plus-terminal cost while enforcing density monotonicity along the predicted trajectory: $s : \mathbb{R}^n \to \mathbb{R}$2 subject to the system dynamics, the CDF inequality above, $s : \mathbb{R}^n \to \mathbb{R}$3, and admissibility constraints (Narayanan et al., 16 Sep 2025).

The paper compares MPC-CDF with MPC-CBF and reports, for a unicycle example, similar solve times but higher minimum distances to obstacles for MPC-CDF across several tunings. For $s : \mathbb{R}^n \to \mathbb{R}$4, the reported average solve times are $s : \mathbb{R}^n \to \mathbb{R}$5, $s : \mathbb{R}^n \to \mathbb{R}$6, and $s : \mathbb{R}^n \to \mathbb{R}$7 seconds, with corresponding minimum distances $s : \mathbb{R}^n \to \mathbb{R}$8, $s : \mathbb{R}^n \to \mathbb{R}$9, and $V$0 meters. For MPC-CBF with $V$1, the times are $V$2, $V$3, and $V$4 seconds, with minimum distances $V$5, $V$6, and $V$7 meters (Narayanan et al., 16 Sep 2025). The paper also reports underwater-vehicle simulations in complex 3D environments, emphasizing larger safety margins and more intuitive tuning for CDFs (Narayanan et al., 16 Sep 2025).

The paper notes that full proofs of stability and recursive feasibility are ongoing and not fully developed there (Narayanan et al., 16 Sep 2025). That caveat is important because it distinguishes demonstrated efficacy from fully closed theoretical treatment.

6. Spatial density functions for coverage control and multi-robot coordination

A different but related tradition uses density functions as task distributions over a workspace. In “Multi-Robot Control Using Time-Varying Density Functions,” the density

$V$8

is a bounded, continuously differentiable time-varying density function encoding the relative importance of points in a domain $V$9 (Lee et al., 2014). For robots at positions $X$0, the locational cost is

$X$1

with Voronoi cells $X$2, cell masses

$X$3

and centroids

$X$4

Critical points satisfy $X$5, yielding centroidal Voronoi tessellations (CVTs) (Lee et al., 2014).

For static densities, Lloyd’s law

$X$6

drives convergence. For time-varying densities, the paper derives the centralized TVD-C algorithm

$X$7

together with distributed Neumann-series approximations such as TVD-D$X$8, where each robot depends only on Voronoi neighbors (Lee et al., 2014). The paper states that TVD-C theoretically guarantees perfect tracking of the time-varying CVT whenever the inverse exists, and that TVD-D$X$9 is always well-posed and performs nearly identically in simulations and robot experiments (Lee et al., 2014).

“Multi-UAV Flood Monitoring via CVT with Gaussian Mixture of Density Functions for Coverage Control” adopts the same CVT backbone but focuses on perceptually updated flood monitoring (Song et al., 22 Oct 2025). There the control density function $u(k)=\kappa(x(k))$0 encodes the spatial importance of points in the monitoring workspace $u(k)=\kappa(x(k))$1. A standard axis-aligned Gaussian model is contrasted with a Gaussian Mixture of Density Functions (GMDF): $u(k)=\kappa(x(k))$2 Here $u(k)=\kappa(x(k))$3 and $u(k)=\kappa(x(k))$4 are obtained from segmented onboard imagery, $u(k)=\kappa(x(k))$5 indicates whether UAV $u(k)=\kappa(x(k))$6 detects flood, and $u(k)=\kappa(x(k))$7 is a small background density (Song et al., 22 Oct 2025).

The induced coverage objective is

$u(k)=\kappa(x(k))$8

with density-weighted centroids

$u(k)=\kappa(x(k))$9

and control law

$s$00

The paper reports consistently higher coverage rates for the GMDF-based formulation than for axis-aligned Gaussian modeling across fleet sizes 16, 20, and 24 in ROS/Gazebo simulations (Song et al., 22 Oct 2025).

In this coverage-control lineage, density functions are not safety certificates or dissipation inequalities. Rather, they are spatial weighting fields that induce optimal placement or motion through CVT mechanics.

7. Related uses, distinctions, and recurrent misconceptions

The literature surrounding “CDFs” also includes several uses of the abbreviation that are not control density functions in the control-theoretic sense. “Neural Likelihoods via Cumulative Distribution Functions” uses CDF to mean cumulative distribution function and develops monotonic neural parameterizations of conditional CDFs for density estimation (Chilinski et al., 2018). “Learning Multivariate CDFs and Copulas using Tensor Factorization” and “A CDF-First Framework for Free-Form Density Estimation” likewise treat CDF as cumulative distribution function, emphasizing density estimation, identifiability, and stable learning targets rather than controller synthesis (Amiridi et al., 2022, Song et al., 26 Mar 2026). These works are relevant only by abbreviation and density-modelling analogy, not by direct control methodology.

A separate nearby theme is shaping state probability density functions in stochastic control. “Lyapunov-based Stochastic Nonlinear Model Predictive Control: Shaping the State Probability Density Functions” uses the Fokker-Planck equation to propagate state PDFs, the Hellinger distance to measure deviation from a reference density, and a stochastic control Lyapunov function to ensure asymptotic stability in probability (Buehler et al., 2015). This is again different from control density functions as occupancy or dissipativity certificates, but it reinforces the broader role of density-valued objects in modern control.

Two misconceptions are therefore common. The first is to treat all “CDF” papers as belonging to the same topic; the evidence does not support that. The second is to equate density-function methods with ordinary barrier methods. In the safe-control papers, density functions are explicitly presented as dual, barrier-like, and convex-amenable alternatives that can avoid some of the bilinearities and non-convexities of barrier-based synthesis, especially when combined with SOS programming and variable substitutions such as $s$01 (Bozdag et al., 21 May 2025, Zheng et al., 2023).

Taken together, the cited work shows that Control Density Functions are best understood as a cluster of density-based control constructions. In stabilization, they generalize control Lyapunov functions through dissipativity (Lazar, 2021). In safety and navigation, they encode occupancy and enable dual formulations of reachability, avoidance, and MPC synthesis (Zheng et al., 2023, Zheng et al., 2023, Bozdag et al., 21 May 2025, Narayanan et al., 16 Sep 2025). In multi-agent coverage, they specify the desired spatial distribution of robots and determine centroidal deployments [(Lee et al., 2014); (Song et al., 22 Oct 2025)]. This suggests that density functions have become a unifying design language for problems where classical state-value or trajectory-value representations are either too rigid or too expensive computationally.