Infinite-Horizon Temporal Anchoring

Updated 17 January 2026

Infinite-horizon temporal anchoring is a paradigm that enforces global objectives by propagating temporal constraints and linking finite computations to infinite trajectories.
It is applied across control, reinforcement learning, economics, and deep models to ensure feasibility and optimality through terminal constraints and state recurrency requirements.
The approach decomposes global specifications into local optimization problems by using anchor constraints that bridge finite windows with long-term performance guarantees.

Infinite-horizon temporal anchoring is a general paradigm for propagating temporal constraints, value functions, or structural properties across an unbounded time axis, ensuring global objectives are satisfied or optimized even when computation, memory, or control is restricted to finite local windows. It manifests as the imposition of anchor constraints, value-function conditions, or state recurrency requirements that link present trajectory segments to future or prescribed terminal behaviors, thereby guaranteeing feasibility or optimality over the entire infinite horizon. The concept is realized across control, reinforcement learning, economics, operator theory, and deep models via domain-specific anchoring mechanisms.

1. Temporal Anchoring in Infinite-Horizon Model Predictive and Temporal Logic Control

Infinite-horizon temporal anchoring is central to receding horizon (model predictive) control schemes for systems with temporal logic constraints. In distributed multi-agent systems executing recurring Signal Temporal Logic (STL) tasks, the infinite-horizon property ψ = □₍₀,∞₎ φ (i.e., φ must hold at every point along an infinite trajectory) is enforced by embedding “STL-anchoring constraints” into each finite-horizon optimization cycle. Specifically, each N-step segment of the trajectory is constrained so that all sliding windows of length N+1 satisfy φ, and the endpoint state x(t_N) lies in the one-step backward-reachable set C₁(x(t)) (i.e., x(t_N) can be controlled back to x(t) in one step):

$x(t_N) \in X_T(x(t)) = \{ x' \mid \exists u: f(x',u) = x(t) \}$

This schedule closes the infinite-horizon “loop” and ensures recursive feasibility: any solution at time t can be shifted forward and extended at t+1. The approach decomposes the global constraint into local agent-level programs, preserving global satisfaction via distributed scheduling (Vlahakis et al., 2023).

A similar principle appears in finite deterministic systems controlled for infinite-horizon Linear Temporal Logic (LTL) properties via receding-horizon optimization. Infinite rewards are “anchored” through a terminal constraint enforcing progress toward recurrent accepting sets (Büchi sets) in the product automaton, implemented via an energy (distance-to-anchor) function V(p):

The receding-horizon controller solves, at each time-step, an optimization maximizing rewards over horizon N, subject to terminal condition V(p_{N|k}) < V(p^*_{N|k-1}) (if not already at the recurrent set), driving the system infinitely often to the anchor (Ding et al., 2012).

This universalizes the anchoring paradigm: by enforcing well-constructed endpoint constraints in each finite optimization, infinite-horizon logical specifications are realized.

2. Terminal Costs and Tail Anchoring in Infinite-Horizon Optimal Control

Temporal anchoring underpins modern approaches to infinite-horizon optimal control and approximate dynamic programming. In constrained nonlinear or stochastic systems, direct infinite-horizon optimization is typically intractable. The canonical workaround is the use of terminal costs or terminal value functions as temporal anchors.

For instance, in nonlinear optimal control, the infinite-horizon problem is regularized using a “finite free final time” transfer to a small terminal set Ω_r containing the equilibrium. The cost-to-go beyond Ω_r is anchored by a local quadratic Lyapunov-type function V_L(x) (typically the solution to an LQR Riccati equation):

$V_r(x) = \min_{u(\cdot), t_f \ge 0} \int_{0}^{t_f} L(x(t), u(t)) dt + V_L(x(t_f)), \quad \dot x(t) = f(x(t), u(t))$

As the radius r → 0, V_r(x) converges uniformly to the true infinite-horizon optimal cost V^*(x). The choice of V_L as a tail anchor allows global infinite-horizon specifications to be satisfied while solving only local finite-horizon problems (Mohamed et al., 2023).

Approximate infinite-horizon predictive control applies this anchoring via a learned parametric value function V_f(x; θ) as the terminal cost in an N-step MPC:

$V_N(x) = \min_{u_0 \dots u_{N-1}} \sum_{k=0}^{N-1} \ell(x_k, u_k) + V_f(x_N; \theta)$

V_f(\cdot;\theta) is trained to satisfy the Bellman equation. The anchoring property is achieved by ensuring that V_f is as close as possible (in Bellman residual sense) to the true tail cost. Performance guarantees explicitly relate infinite-horizon suboptimality and required horizon N to this residual (Beckenbach et al., 2021).

3. Temporal Anchoring in Stochastic and Time-Inconsistent Control

In time-inconsistent stochastic optimal control, infinite-horizon temporal anchoring provides the key technical device enabling well-posed equilibrium constructions. When discounting is non-exponential (e.g., hyperbolic), the classical dynamic programming principle fails. Temporal anchoring is introduced by assuming that after a sufficiently large time T₀, preferences revert to exponential (time-consistent) discounting:

$\beta(r) = e^{-\delta r}, \quad \forall r \geq T_0$

This partitions the problem into a near-horizon (finite, time-inconsistent, solved via equilibrium-HJB equations with the distant-value anchor as terminal condition) and a far-horizon (infinite, time-consistent, solvable via traditional HJB). The coupling at T₀ temporally anchors the infinite tail, bridging fundamentally incompatible preference regimes and guaranteeing the existence and uniqueness of equilibria (Wei et al., 18 Sep 2025).

4. Anchoring via Continuation and Foresight in Large-Scale Dynamic Systems

Economic models with heterogeneous agents and aggregate shocks manifest infinite-horizon temporal anchoring via the N-bounded foresight equilibrium (N-BFE) framework. Here, agents optimize expected utility over an infinite horizon, but forecasts of the cross-sectional population state s_t are only accurate for the next N periods. Thereafter, all trajectories are anchored to a fixed continuation value s^C. Formally, agent value is computed as:

$V^{\mathcal I(t,N,z_t)}(x_t, z_t, s_t) = \sup_{\{a_\tau\}} E_t^{\mathcal I} \Big[ \sum_{\tau=t}^{t+N-1} \beta^{\tau-t} \pi(x_\tau, z_\tau, a_\tau, s_\tau) + \beta^N V^c(x_{t+N}, s_{t+N}) \Big]$

The “temporal anchoring” at s^C ensures tractability and equilibrium uniqueness, enabling analysis of forecast errors, volatility, and agent memory in high-dimensional dynamic economies (Islah et al., 23 Feb 2025).

5. Temporal Anchoring in Infinite-Horizon Reinforcement Learning and Generative Models

Infinite-horizon temporal anchoring is foundational in reinforcement learning, both in value function estimation and generative modeling. Value-based RL uses the Bellman fixed-point equation as the infinite-horizon anchor: the value at each state is defined recursively in terms of the values of successor states, with discount γ ensuring convergence. Temporal-difference (TD) algorithms enforce the Bellman residual to be zero in expectation, thus temporally anchoring predictions to the infinite-horizon optimum:

$V^*(s) = \max_{a} \mathbb{E}[ R_{t+1} + \gamma V^*(S_{t+1}) \mid S_t = s, A_t = a ]$

In generative models for infinite-horizon prediction, the γ-model estimates the discounted-occupancy distribution over states by enforcing a Bellman-like recursion in the generative network:

This generative enforcement is the statistical analogue of temporal anchoring: the global infinite-horizon distribution is recursively defined and anchored at every local prediction (Janner et al., 2020). Similar principles underlie the estimation of infinite-horizon dynamic treatment regimes using TD-residuals (Ertefaie, 2014).

6. Temporal Anchoring with Asymptotic State and Operator-Theoretic Constraints

In functional analytic infinite-horizon control with asymptotic state constraints, anchoring is realized via constraints on the limiting behavior of the trajectory, i.e., requiring $\lim_{k \to \infty} x_k = x_\infty$ . Optimality conditions (weak and strong Pontryagin principles) are developed in Banach sequence spaces, with the anchoring constraint yielding boundary (transversality) conditions on the Lagrange multipliers (costate):

$\lim_{k \to \infty} p_k = 0$

This acts as a “temporal anchor” for the infinite sequence, ensuring the state approaches the prescribed limit and enabling strong sufficiency results for the infinite-horizon problem (Blot et al., 2015).

A generalization appears in operator-theoretic treatments of deep or embedding space dynamics, where temporal anchoring is formalized as convergence to a unique intersection of nested affine projection sets after repeated drift–projection compositions. Explicit contraction envelopes and robustness to perturbations ensure that block-wise application of drift maps and event-indexed affine projections anchor the sequence to a unique limit in Hilbert space (Alpay et al., 13 Aug 2025).

7. Temporal Anchoring in Long-Form Deep Generative Models

In large-scale generative (e.g., video diffusion) models, infinite-horizon temporal anchoring is implemented at the architectural level to allow unbounded autoregressive generation. For example, Infinity-RoPE removes fixed temporal position limits in rotary positional embedding via block-relativistic schemes that turn the absolute time axis into a local moving frame (relativistic remapping of positions). Combined with inference-time “flushing” (re-anchoring semantics by refreshing short-term memory at prompt transitions) and scene-cut operators (explicit index discontinuities), this enables models to anchor temporal geometry for arbitrarily long rollouts, while preserving local prompt-conditioning and semantic coherence across the infinite horizon (Yesiltepe et al., 25 Nov 2025).

References: