Persistent Uncertainty in Control & Modeling

• Persistent uncertainty is the evolving state of indecision that persists, accumulates, and regenerates over time in dynamic systems.
• It appears across fields such as surveillance, language model reasoning, and macroeconomics, where it influences control, failure prediction, and volatility analysis.
• Applications utilize methods like optimal control, IPA, threshold-based policies, and invariant set computation to manage and mitigate persistent uncertainty.

Searching arXiv for papers on "persistent uncertainty" and closely related uses across control, monitoring, and LLMs. Persistent uncertainty is a class of dynamical uncertainty phenomena in which uncertainty is not treated as a one-shot confidence score but as a state that evolves, persists, and often regrows over time. In the literature, the term appears in several technically distinct settings: as an uncertainty state attached to targets, regions, or queues in persistent monitoring and surveillance; as a failure signature in chain-of-thought traces of LLMs; as a formally typed modeling attribute in uncertainty-aware systems engineering; and as an autoregressive macroeconomic series extracted from real-time data revisions (Cassandras et al., 2012, Thoria et al., 4 Jun 2026, Zhang et al., 25 Feb 2026, Takakura et al., 2020). A common structural theme is temporal carryover: uncertainty accumulates when unattended, decreases under observation or intervention, and therefore induces optimization, certification, or diagnosis problems over long or infinite horizons.

1. Core definitions and recurrent state-space forms

In persistent monitoring and surveillance, uncertainty is usually represented as a nonnegative state variable coupled to agent motion and sensing. In one-dimensional persistent monitoring, the uncertainty at sampling point $i$ is $R_i(t)$, with dynamics

$$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$

where $A_i>0$ is an inflow rate, $B>A_i$ is the maximal decay rate, and $P_i(s)$ is the joint detection probability induced by the agent positions (Cassandras et al., 2012). On graphs, the same idea appears as a virtual buffer or “uncertainty queue,” with

$\dot R_i(t)=A_i-B_i\,N_i(t)$

for almost all $t$, subject to $R_i(t)\ge 0$, where $N_i(t)$ counts how many agents are at node $R_i(t)$0 (Zhou et al., 2018). In persistent surveillance by autonomous robots, region $R_i(t)$1 carries a discrete uncertainty state $R_i(t)$2 satisfying

$R_i(t)$3

with $R_i(t)$4 when the region is observed and $R_i(t)$5 otherwise (Nenchev et al., 7 May 2026).

These formulations share an explicit asymmetry between observation and neglect. Observation can drive uncertainty downward, sometimes to zero; absence of observation lets it regrow. This suggests that persistence is not merely long duration but repeated regeneration under partial coverage, resource limits, or motion constraints.

A different use of the term appears in language-model failure analysis. There, persistent uncertainty is defined through a prefix diagnostic curve. For a reasoning trace $R_i(t)$6, a classifier is trained on prefix-aggregated token-level uncertainty features $R_i(t)$7 to predict whether the final answer is incorrect, and

$R_i(t)$8

A configuration exhibits persistent uncertainty iff $R_i(t)$9 is non-decreasing in $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$0 and $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$1, so that no early prefix outperforms the full trace (Thoria et al., 4 Jun 2026).

2. Continuous-space persistent monitoring as optimal control

The classical persistent monitoring problem is posed as an optimal-control problem over agent trajectories and uncertainty states. In the one-dimensional formulation, $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$2 agents move in a mission space $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$3 or $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$4, with dynamics $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$5, $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$6, and objective

$$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$7

The sensing model is distance-dependent, and the joint detection probability at point $$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$8 is

$$\dot R_i(t)= \begin{cases} 0, & R_i=0 \text{ and } A_i-B\,P_i(s)\le 0\ A_i-B\,P_i(s), & \text{otherwise,} \end{cases}$$9

The resulting Hamiltonian analysis yields bang-bang motion with dwell arcs: each optimal trajectory consists of full-speed motion segments $A_i>0$0 punctuated by waiting intervals $A_i>0$1, and the problem reduces to optimizing switching locations $A_i>0$2 and dwell times $A_i>0$3 (Cassandras et al., 2012).

The reduction to a finite-dimensional parameterization is operationally important because it enables Infinitesimal Perturbation Analysis (IPA). IPA computes derivatives of $A_i>0$4 with respect to switching locations and dwell times from a single hybrid-system sample path, using endogenous event-time sensitivities and inter-event state sensitivities. The same work also proposes a receding-horizon controller that solves a shorter local problem over horizon $A_i>0$5, applies the control over a smaller interval $A_i>0$6, and repeats online (Cassandras et al., 2012).

A two-dimensional infinite-horizon variant formulates persistent uncertainty around static point targets with finite sensing radii. There, the agent seeks a steady-state minimum-time periodic trajectory over which each target uncertainty is driven to zero during each visit. The paper decomposes the problem into local optimal-control subproblems, coupled by boundary conditions on target-entry and target-exit points, and solves it through an online bilevel optimization scheme: the lower level solves draining and switching optimal-control problems, while the upper level updates the boundary angles via gradient descent using dual information. Under the stated assumptions—constant $A_i>0$7, $A_i>0$8, non-overlapping sensing disks, and existence of a periodic cycle—the hybrid draining problem admits an exact smooth reformulation (Theorem 2), local subproblems are solved by direct multiple shooting in $A_i>0$9–$B>A_i$0 s, typical cycle updates take $B>A_i$1 s, and cycle time is reduced by up to $B>A_i$2 relative to a greedy “go to next target, dwell until $B>A_i$3” baseline (Hall et al., 2023).

In both formulations, persistent uncertainty is the source of temporal coupling. A decision that is locally efficient can be globally poor because unattended uncertainty states continue to grow elsewhere.

3. Graph-based, threshold-based, and mobile-target formulations

On graphs, persistent uncertainty is attached to nodes rather than spatial sampling points. A team of agents moves on a graph $B>A_i$4, with direct travel allowed only along edges, and minimizes

$B>A_i$5

Because exact dynamic programming is generally intractable, a distributed threshold-based policy class is introduced. For each agent $B>A_i$6, a matrix $B>A_i$7 specifies self-thresholds $B>A_i$8 and neighbor thresholds $B>A_i$9. While an agent is at node $P_i(s)$0, it stays if $P_i(s)$1 or if all neighbor uncertainties remain below their thresholds; otherwise it departs to the first neighbor whose threshold is crossed. The closed loop is a hybrid automaton, enabling IPA-based online gradient descent on the thresholds (Zhou et al., 2018).

A central structural result is specific to the single-agent case: for fixed visiting sequence, the IPA gradient with respect to self-thresholds is strictly positive, so gradient descent drives $P_i(s)$2. The corresponding policy is “stay at any node $P_i(s)$3 until $P_i(s)$4, then move on.” The multi-agent case is qualitatively different. The paper gives a two-agent, five-target example in which some optimal self-thresholds remain positive, showing that shared service breaks the single-agent zero-threshold structure (Zhou et al., 2018).

A related but more general one-dimensional problem considers mobile targets $P_i(s)$5 with known velocities and uncertainty states $P_i(s)$6. The sensing probability is

$P_i(s)$7

and the joint detection probability is

$P_i(s)$8

Under the assumptions that target speeds do not exceed $P_i(s)$9 and no two targets ever lie within $\dot R_i(t)=A_i-B_i\,N_i(t)$0 of each other, the optimal-control problem admits a finite-dimensional representation. Any admissible control can be replaced by a hybrid “bang-bang plus track” control that never does worse, and an optimal finite-dimensional control exists. Two parameterizations are then developed: an “optimal” one based on switching locations and convex combinations of target velocities, and a “practical” one using only target positions through a clipped PI controller. Event-based IPA again supplies the gradient for parameter updates (Hall et al., 2022).

These graph and mobile-target variants make explicit that persistent uncertainty is often less about a particular geometry than about a hybrid switching structure: motion, service, and uncertainty dynamics interact through event times.

4. Persistent feasibility, adjustable uncertainty sets, and invariant monitors

In robust control, persistent uncertainty appears when the uncertainty set itself becomes a decision variable. Robust Model Predictive Control with Adjustable Uncertainty Sets (RMPC-AU) considers the discrete-time system

$\dot R_i(t)=A_i-B_i\,N_i(t)$1

with state and input constraints and disturbances $\dot R_i(t)=A_i-B_i\,N_i(t)$2, where $\dot R_i(t)=A_i-B_i\,N_i(t)$3 is optimized jointly with the control policy. The finite-horizon objective trades nominal cost against a measure $\dot R_i(t)=A_i-B_i\,N_i(t)$4 of uncertainty-set size, while persistent feasibility is guaranteed by a terminal condition on the pair $\dot R_i(t)=A_i-B_i\,N_i(t)$5. The paper introduces adjustable control invariant sets and adjustable positive invariant sets in the product space $\dot R_i(t)=A_i-B_i\,N_i(t)$6, proves that an adjustable invariant terminal set is sufficient for recursive feasibility, and computes the largest invariant set via the fixed-point iteration

$\dot R_i(t)=A_i-B_i\,N_i(t)$7

until convergence (Kim et al., 2018).

The same invariance logic is repurposed for runtime monitoring of black-box autonomous surveillance. The workspace is partitioned into finitely many parts $\dot R_i(t)=A_i-B_i\,N_i(t)$8, each with uncertainty state $\dot R_i(t)=A_i-B_i\,N_i(t)$9, while robot motion obeys $t$0. The full closed loop is modeled as a state-dependent hybrid system with linear parameter-varying dynamics. Offline, one seeks a robust positively invariant set $t$1, equivalently the maximal fixed point of

$t$2

Because direct computation grows exponentially in the number of uncertainty regions, the paper computes per-part maximal RPIs $t$3 and, under the assumptions in Proposition 1, reconstructs the full invariant as

$t$4

Online monitoring reduces to checking each half-space membership condition and declaring “healthy” iff all checks pass. In a 10-cell labyrinth patrol with a real robot, the compositional monitor ran in $t$5 ms/step, raised no false alarms, and provided positive lead-time—several seconds—before any $t$6 reached its critical threshold (Nenchev et al., 7 May 2026).

A related multi-robot persistence problem adds energy feasibility to uncertainty management. In mEclares, uncertainty in a stochastic spatiotemporal field is represented through “clarity”

$t$7

and, for a scalar cell under observation model $t$8, the clarity dynamics satisfy

$t$9

Clarity is converted into a target information spatial distribution (TISD) for ergodic exploration, while persistent operation is guaranteed by the Robustmesch scheduler. Under Lemma 1, at most

$R_i(t)\ge 0$0

robots can be supported while maintaining return gaps, and Theorem 1 gives recursive feasibility of minimum-energy and minimum-gap conditions under the scheduling and fail-safe rules. Hardware experiments with four quadrotors and a mobile ground charger reported ROS 2 round-trip latency of $R_i(t)\ge 0$1 ms, below $R_i(t)\ge 0$2 s, with no unintended fallbacks (Naveed et al., 16 May 2025).

5. Persistent uncertainty in language-model reasoning

In chain-of-thought analysis, persistent uncertainty is one of two empirically observed failure modes, the other being committed failure. The diagnostic machinery is token-level. At generation step $R_i(t)\ge 0$3, the model yields a distribution $R_i(t)\ge 0$4, from which one computes entropy,

$R_i(t)\ge 0$5

margin $R_i(t)\ge 0$6, negative log-likelihood $R_i(t)\ge 0$7, nucleus size $R_i(t)\ge 0$8, and near-tie count $R_i(t)\ge 0$9. Mean and maximum aggregations over prefix windows $N_i(t)$0 define features $N_i(t)$1, and a classifier predicts whether the final answer is wrong from these prefix statistics (Thoria et al., 4 Jun 2026).

Persistent uncertainty is then the regime in which the classifier’s $N_i(t)$2 rises monotonically with prefix length and is maximized only by the full trace. In the reported experiments, 23 model-dataset configurations were tested, with an applicability band that left 14 committed and 9 persistent configurations. One persistent case is Llama3.1-8B on MATH-500, where $N_i(t)$3, $N_i(t)$4, and $N_i(t)$5. Across all 23 configurations, a joint sign-test on the sign of $N_i(t)$6 in committed versus persistent regimes rejected uniformity at $N_i(t)$7 (Thoria et al., 4 Jun 2026).

A more structural argument about uncertainty in transformers is developed in the distinction between descriptive and regulatory uncertainty. Descriptive uncertainty reports properties of the model’s internal distribution—such as Shannon entropy—without feeding back into energy expenditure, control signals, or model parameters at inference. Regulatory uncertainty, by contrast, participates in adaptation or decision costs and requires an error-dependent penalty: $N_i(t)$8 Under fixed weights $N_i(t)$9, fixed inference temperature $R_i(t)$00, and no auxiliary control pathway, the paper’s Softmax Decoupling Theorem shows that token entropy

$R_i(t)$01

is a deterministic function of logits and $R_i(t)$02, and therefore purely descriptive at inference. Empirically, across three locally deployed LLMs—3B, 8B, and 70B parameters—token-level Shannon entropy was statistically invariant across pattern retrieval, causal operator application, and out-of-distribution causal generalization tasks, with all pairwise $R_i(t)$03 and within-model entropy ranges of $R_i(t)$04-$R_i(t)$05 nats, while task accuracy ranged from $R_i(t)$06 to $R_i(t)$07. The reported conclusion is that entropy and accuracy are orthogonal, and that this decoupling is scale-invariant: larger models achieve higher accuracy but preserve entropy flatness across task categories (Eldin, 17 May 2026).

A common misconception in LLM practice is that token-level confidence directly tracks reasoning quality. The reported evidence does not support that claim: one line of work finds persistent uncertainty only in the temporal pattern of failure detectability across traces, and another finds that token entropy itself is structurally decoupled from correctness under standard transformer inference.

6. Formal representation and other quantitative uses

In model-based systems engineering, uncertainty is formalized directly as a first-class modeling object. The PSUM-SysML v2 extension maps the PSUM metamodel into SysML v2 using stereotypes such as «BeliefStatement», «IndeterminacySource», «IndeterminacySpecification», «Uncertainty», «UncertaintyTopic», and «Effect». An uncertainty attachment on a belief statement $R_i(t)$08 is defined as

$R_i(t)$09

where $R_i(t)$10. The profile also gives propagation and composition rules. If $R_i(t)$11 affects $R_i(t)$12, then

$R_i(t)$13

and if independent uncertainties $R_i(t)$14 jointly guard a transition, then

$R_i(t)$15

The extension was validated through seven case studies and is intended to support explicit specification of indeterminacy sources and consistent uncertainty propagation analyses (Zhang et al., 25 Feb 2026).

In macroeconomics, persistent uncertainty is quantified as a latent volatility series inferred from real-time GDP data revisions. Let $R_i(t)$16 denote the conditional standard deviation of the news innovations in the first and last vintages; the paper models its persistence through

$R_i(t)$17

Typical estimates across OECD countries are $R_i(t)$18–$R_i(t)$19, implying half-lives of $R_i(t)$20–$R_i(t)$21 quarters. In a panel VAR, the effects of uncertainty shocks are stronger and more persistent in countries with low employment protection than in countries with high employment protection; the reported output half-life is about $R_i(t)$22 quarters in low-EPL countries and about $R_i(t)$23–$R_i(t)$24 quarters in high-EPL countries (Takakura et al., 2020).

Taken together, these formalisms indicate that persistent uncertainty is best understood as a temporal property of systems in which uncertainty survives local correction and therefore enters optimization, certification, or diagnosis over extended horizons. In control and surveillance, it is a state to be reduced while preserving feasibility. In LLM reasoning, it is a trace-level signature of failures whose evidence accumulates across tokens. In systems engineering, it is an explicitly typed attribute with propagation semantics. In macroeconomics, it is a measurable stochastic process with nontrivial half-life. The shared technical content is not a single equation but a recurring architecture: uncertainty evolves, couples across time, and cannot be analyzed adequately as an isolated instantaneous quantity.