--- title: Persistent Uncertainty in Control & Modeling url: https://www.emergentmind.com/topics/persistent-uncertainty type: topic --- # Persistent Uncertainty in Control & Modeling Searching arXiv for recent papers on "persistent uncertainty" and closely related uses across control, monitoring, and language models. 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 language models; as a formally typed modeling attribute in uncertainty-aware systems engineering; and as an autoregressive macroeconomic series extracted from real-time data revisions [1202.6110] [2606.06635] [2602.21641] [2006.02092]. 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 [1202.6110]. 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 \(i\) [1803.02798]. In persistent surveillance by autonomous robots, region \(i\) carries a discrete uncertainty state \(z_{i,k}\) satisfying \[ z_{i,k+1}=\gamma_i(x_k)z_{i,k}, \] with \(\gamma_i(x)\in[a_i^{\min},a_i^{\max}]\subset(0,1)\) when the region is observed and \(\gamma_i(x)\in[b_i^{\min},b_i^{\max}]\subset(1,\infty)\) otherwise [2605.06062]. 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 \(x_1,\dots,x_L\), a classifier is trained on prefix-aggregated token-level uncertainty features \(\phi_T\) to predict whether the final answer is incorrect, and \[ \Delta(T)=\mathrm{PR\text{-}AUC}(T)-\mathrm{PR\text{-}AUC}(\mathrm{full}). \] A configuration exhibits persistent uncertainty iff \(\mathrm{PR\text{-}AUC}(T)\) is non-decreasing in \(T\) and \(\max_{T0\), \(B_i>A_i\), 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 \(\sim 0.01\)–\(0.05\) s, typical cycle updates take \(\sim 0.5\) s, and cycle time is reduced by up to \(20\%\) relative to a greedy “go to next target, dwell until \(R=0\)” baseline [2304.03667]. 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 \(G=(V,E)\), with direct travel allowed only along edges, and minimizes \[ J=\frac{1}{T}\int_0^T \sum_{i=1}^M R_i(t)\,dt. \] Because exact dynamic programming is generally intractable, a distributed threshold-based policy class is introduced. For each agent \(a\), a matrix \(\Theta^a=[\theta_{ij}^a]\) specifies self-thresholds \(\theta_{ii}^a\) and neighbor thresholds \(\theta_{ij}^a\). While an agent is at node \(i\), it stays if \(R_i(t)>\theta_{ii}^a\) 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 [1803.02798]. 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 \(\theta_i\to 0\). The corresponding policy is “stay at any node \(i\) until \(R_i(t)=0\), 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 [1803.02798]. A related but more general one-dimensional problem considers mobile targets \(\xi_i(t)\) with known velocities and uncertainty states \(R_i(t)\ge 0\). The sensing probability is \[ p_{ij}(\xi_i(t),s_j(t))=\max\left\{0,1-\frac{|\xi_i(t)-s_j(t)|}{r_j}\right\}, \] and the joint detection probability is \[ P_i(t)=1-\prod_{j=1}^N [1-p_{ij}(\xi_i(t),s_j(t))]. \] Under the assumptions that target speeds do not exceed \(1\) and no two targets ever lie within \(2\max_j r_j\) 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 [2210.01294]. 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 \[ x_{k+1}=Ax_k+Bu_k+Ew_k, \] with state and input constraints and disturbances \(w_k\in \mathbb W_k\), where \(\mathbb W_k\) is optimized jointly with the control policy. The finite-horizon objective trades nominal cost against a measure \(\rho(\mathbb W)\) of uncertainty-set size, while persistent feasibility is guaranteed by a terminal condition on the pair \((x_{N|t},\mathbb W_{N-1|t})\). The paper introduces adjustable control invariant sets and adjustable positive invariant sets in the product space \(\mathbb X\times \mathcal P(\mathbb R^{n_w})\), proves that an adjustable invariant terminal set is sufficient for recursive feasibility, and computes the largest invariant set via the fixed-point iteration \[ \Omega_{k+1}=\mathrm{Pre}(\Omega_k,\mathbb S)\cap \Omega_k \] until convergence [1809.06540]. The same invariance logic is repurposed for runtime monitoring of black-box autonomous surveillance. The workspace is partitioned into finitely many parts \(P_i\), each with uncertainty state \(z_{i,k}\in[0,z_i^{\mathrm{crit}}]\), while robot motion obeys \(x_{k+1}=x_k+T_su_k\). 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 \(\mathcal S\subset D\), equivalently the maximal fixed point of \[ T(\mathcal S)=D\cap \mathrm{Pre}(\mathcal S). \] Because direct computation grows exponentially in the number of uncertainty regions, the paper computes per-part maximal RPIs \(S_i^{\max}\) and, under the assumptions in Proposition 1, reconstructs the full invariant as \[ \mathcal S=\{(x,z,u)\in D\mid (x,z_i,u)\in S_i^{\max}\ \forall i\}. \] 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 \(\approx 0.3\) ms/step, raised no false alarms, and provided positive lead-time—several seconds—before any \(z_i\) reached its critical threshold [2605.06062]. A related multi-robot persistence problem adds energy feasibility to uncertainty management. In mEclares, uncertainty in a stochastic spatiotemporal field is represented through “clarity” \[ q[X]=\left(1+\frac{e^{2h[X]}}{(2\pi e)^n}\right)^{-1}\in[0,1], \] and, for a scalar cell under observation model \(y=C(x)m+v\), the clarity dynamics satisfy \[ \dot q=\frac{C(x)^2}{R}(1-q)^2-Qq^2. \] 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 \[ N^*=1+\left\lfloor\frac{T_{F,0}^{1'}-T_R-T_E}{T_\delta}\right\rfloor \] 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 \(\approx 600\pm 150\) ms, below \(T_E=1.5\) s, with no unintended fallbacks [2505.11663]. ## 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 \(t\), the model yields a distribution \(p^{(t)}\), from which one computes entropy, \[ \mathcal H_t=-\sum_i p_i^{(t)}\log p_i^{(t)}, \] margin \(\mathcal M_t=p^{(t)}_{(1)}-p^{(t)}_{(2)}\), negative log-likelihood \(\mathcal L_t=-\log p^{(t)}_{(1)}\), nucleus size \(\mathcal N_t\), and near-tie count \(\mathcal T_t\). Mean and maximum aggregations over prefix windows \(W_T\) define features \(\phi_T\), and a classifier predicts whether the final answer is wrong from these prefix statistics [2606.06635]. Persistent uncertainty is then the regime in which the classifier’s \(\mathrm{PR\text{-}AUC}(T)\) 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 \(\mathrm{PR\text{-}AUC}@128=0.7598\), \(\mathrm{PR\text{-}AUC}@2048=0.8163\), and \(\mathrm{PR\text{-}AUC}(\mathrm{full})=0.8212\). Across all 23 configurations, a joint sign-test on the sign of \(\Delta\) in committed versus persistent regimes rejected uniformity at \(p<10^{-4}\) [2606.06635]. 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: \[ \frac{\partial E_{\mathrm{cost}}}{\partial \epsilon}>0, \qquad \epsilon=|\hat y-y^*|. \] Under fixed weights \(\theta\), fixed inference temperature \(\tau\), and no auxiliary control pathway, the paper’s Softmax Decoupling Theorem shows that token entropy \[ H_t=-\sum_i P_\theta(v_i\mid ctx_t)\log P_\theta(v_i\mid ctx_t) \] is a deterministic function of logits and \(\tau\), and therefore purely descriptive at inference. Empirically, across three locally deployed language models—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 \(p\ge 0.568\) and within-model entropy ranges of \(0.011\)-\(0.028\) nats, while task accuracy ranged from \(0\%\) to \(100\%\). 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 [2605.18909]. 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 \(M\) is defined as \[ U=\bigl\langle M;\;kind,\;nature,\;perspective,\;pattern,\;reducibility,\;S,\;E\bigr\rangle, \] where \(pattern\in\{\text{periodic, persistent, sporadic, transient, random}\}\). The profile also gives propagation and composition rules. If \(M_1\) affects \(M_2\), then \[ P(M_2 \text{ fails})= P(M_2\mid M_1 \text{ fails})P(M_1 \text{ fails})+ P(M_2\mid M_1 \text{ works})P(M_1 \text{ works}), \] and if independent uncertainties \(U_1,\dots,U_k\) jointly guard a transition, then \[ P_{\rm joint}=1-\prod_{i=1}^k (1-P(U_i)). \] The extension was validated through seven case studies and is intended to support explicit specification of indeterminacy sources and consistent uncertainty propagation analyses [2602.21641]. In macroeconomics, persistent uncertainty is quantified as a latent volatility series inferred from real-time GDP data revisions. Let \(U_t\) denote the conditional standard deviation of the news innovations in the first and last vintages; the paper models its persistence through \[ U_t=\rho_U U_{t-1}+u_t, \qquad t_{1/2}=\frac{\ln(0.5)}{\ln(\rho_U)}. \] Typical estimates across OECD countries are \(\rho_U\approx 0.90\)–\(0.95\), implying half-lives of \(5\)–\(10\) 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 \(8\) quarters in low-EPL countries and about \(4\)–\(5\) quarters in high-EPL countries [2006.02092]. 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. Source: https://www.emergentmind.com/topics/persistent-uncertainty