Finite-Horizon Framework: Control & Optimization

• The finite-horizon framework is a model that confines decision-making to a fixed time interval, enabling non-stationary policies and terminal cost formulations.
• It employs backward induction and dynamic programming to yield explicit value function recursions and unique solutions across control, reinforcement learning, and game theory.
• The framework facilitates approximating infinite-horizon problems and supports efficient policy synthesis in applications like receding horizon control and risk-sensitive decision-making.

A finite-horizon framework is a mathematical, algorithmic, or control-theoretic approach in which optimization or decision-making is explicitly constrained to a fixed time (or stage, or episode) length. This paradigm appears broadly across control theory, reinforcement learning, game theory, behavioral systems, and operations research, with each domain formalizing and exploiting the “finite-horizon” in distinct ways. The core idea is to replace, approximate, or constrain possibly intractable or ill-posed infinite-horizon problems by those considered over a specified, finite number of steps, typically denoted by $T$, $N$, or $H$. This finite constraint fundamentally shapes both the structure of optimal policies and the complexity of algorithms and equilibrium concepts.

1. Formulation and Role in Dynamic Optimization

A defining property of the finite-horizon framework is its structuring of problems—control, planning, reinforcement learning, or dynamic games—over an explicit, bounded time interval, $[0,T]$ in continuous time or steps $0, 1, ..., T$ in discrete time. The solution concept, whether it be an optimal policy, trajectory, or strategy profile, is generally rendered non-stationary: controls or actions depend on both the current state and the remaining time-to-go, in contrast to the typically stationary nature of infinite-horizon formulations (VP et al., 2021, Guin et al., 2022, Rozada et al., 17 Jan 2025). The finite horizon fundamentally alters the value function recursions—backward induction or dynamic programming equations now terminate at a specific terminal cost or state, and the nature of future uncertainty is correspondingly finite.

In continuous-time systems, such as risk-sensitive CTMDPs, the cost is given as an exponential of integrated costs over $[0,T]$ plus a terminal reward, requiring the backward-in-time solution of a time-dependent Hamilton–Jacobi–BeLLMan (HJB) or optimality equation (Guo et al., 2018). In optimal control, the Hamilton–Jacobi–BeLLMan–Type (HJBT) equation admits a unique solution when posed on a finite horizon, obviating the ambiguity or multiplicity of steady-state HJB solutions seen in infinite-horizon settings (Fotiadis et al., 28 May 2025).

2. Implications for Policy, Value Functions, and Equilibria

Finite-horizon settings induce structural and computational differences compared to their infinite-horizon counterparts:

• Non-stationary policies: In Markov Decision Processes (MDPs) and dynamic games, the optimal policy at stage $t$ explicitly depends on $t$ as well as the state, with each stage typically parameterized by its own decision variable or policy vector (Guin et al., 2022, Chen et al., 25 Sep 2024, Rajasekaran et al., 2021, Huang et al., 24 Jun 2025).
• Distinct value function/recursion structure: Backward induction proceeds from a stage-dependent or explicitly time-varying terminal value, as in $V_T(s)$ for the final stage, with BeLLMan or Riccati equations terminating at $T$ (VP et al., 2021, Rozada et al., 17 Jan 2025, Huang et al., 24 Jun 2025).
• Unique solutions and fixed points: Finite-horizon formulations typically admit unique solutions for policies or feedback matrices due to the finite backward recursion, unlike infinite-horizon games where multiple equilibria or even periodic (cyclic) Nash equilibria can arise (Salizzoni et al., 28 Aug 2025).
• Approximation of infinite-horizon behavior: For sufficiently large $T$, finite-horizon solutions can approximate infinite-horizon optima. Papers rigorously quantify the error between finite- and infinite-horizon solutions, showing decay as $T\to\infty$, and often provide explicit bounds in terms of the “distance” between the strategies or value functions (Huang et al., 24 Jun 2025, Sayin, 2023).

3. Algorithmic Frameworks and Controller Synthesis

Finite-horizon frameworks underpin several key algorithmic paradigms and synthesis methodologies:

• Receding horizon (model predictive) control: At each step, an optimal trajectory over a finite horizon is computed, immediate action taken, then the process repeated. Terminal/energy constraints enforce progress toward high-level specifications—for instance, ensuring satisfaction of LTL formulas in automata-based synthesis (Ding et al., 2012).
• Counter-based and memory-bounded strategies: For tasks such as reaching a target state within $T$ steps in stochastic games, optimal or nearly optimal strategies require memory logarithmic in both precision $1/\epsilon$ and in $T$. The non-stationarity and need to “count-to-$T$” fundamentally increase memory requirements over the infinite-horizon memoryless case (Chatterjee et al., 2012).
• Actor-critic and policy gradient methods: Algorithms for finite-horizon MDPs parameterize policies (and critics) by stage, yielding architectures and convergence that must contend with the explicit time-varying nature of optimal solutions. Learning algorithms are structured to address constraints, duality, or multi-agent equilibria within the finite window (Guin et al., 2022, VP et al., 2021, Sayin, 2023).
• Optimization under finite budgets: In online optimization and numerical methods, a “finite-horizon optimization” framework seeks the best configuration or acceleration over a prescribed iteration budget, frequently requiring reformulation to expose hidden convexity or efficient parametrization subject to iteration or computational constraints (Zhang et al., 30 Dec 2024).

4. Applications in Games, Planning, and Control Synthesis

Finite-horizon frameworks support a range of high-level objectives:

• Temporal logic control: Satisfying complex specifications (e.g., always eventually visit target regions and always avoid unsafe regions) is achieved by converting the task to automaton-based trajectory planning, with the finite horizon ensuring recursive feasibility and satisfaction of the specification via terminal constraints on an “energy” function (Ding et al., 2012).
• Constrained or risk-sensitive settings: Temporal constraints are essential when modeling participation- or commitment-constrained contracts, where time-limited opportunities (or the impossibility of threats beyond $T$) fundamentally shape the optimal design and induce “ratcheting” behavior in optimal contracts (Jeon et al., 2018). Finite-horizon risk-sensitive MDPs require unique solutions to nonlinear optimality equations even with unbounded rates (Guo et al., 2018).
• Studies of equilibrium structure: In multi-agent linear-quadratic games, the finite-horizon Nash equilibrium is unique and governed by coupled backward Riccati recursions. These games can exhibit convergence to infinite-horizon equilibria, periodic orbits, or, for some terminal cost choices, non-convergent bounded behaviors, illuminating the link between terminal conditions and long-term asymptotics (Huang et al., 24 Jun 2025, Salizzoni et al., 28 Aug 2025).
• Reinforcement learning and dynamic programming: Finite-horizon Q-learning generalizes dynamic programming to non-stationary settings, with convergence and stability proved via ODE methods, and finds direct real-world application in smart grid energy management and battery scheduling (VP et al., 2021, Rozada et al., 17 Jan 2025).

5. Structural and Metric Properties in System Identification

Finite-horizon frameworks enable fine-grained, invariant comparisons across linear (and LTI) behaviors in system identification and anomaly detection:

• Grassmannian geometry and minimum distance modeling: By considering the “subspaces” associated with system trajectories over finite time intervals, new classes of metrics—parameterized by principal angles and penalizing both misfit and complexity—emerge (Padoan et al., 28 Mar 2025). These metrics exhibit invariance to coordinate changes, rotations, and input-output partition permutations, resolving ambiguity in model selection and anomaly identification in time series data.
• Complexity–misfit trade-offs: The trade-off between trajectory misfit and model complexity is naturally encoded via non-Euclidean distances, enabling geometric optimization for most powerful unfalsified models (MPUM) and principled model selection.

6. Approximation, Error Bounds, and Scaling in High Dimensions

A unifying feature of finite-horizon frameworks is quantitative control of the approximation error as a function of the horizon $T$:

• Episodic equilibria and error decay: In stochastic games, episodic equilibrium strategies parameterized by state and time step achieve bounded errors with respect to infinite-horizon objectives, with error bounds decreasing as episode length increases. Explicit formulas relate $\epsilon$-accuracy to horizon length, time-averaging, or discount factors (Sayin, 2023).
• Sample complexity in RL: For policy gradient and learning in finite-horizon MDPs with general state/action spaces, analyses show that every stationary point is globally optimal under the Kurdyka–Łojasiewicz (KL) condition, and sample complexity for $\epsilon$-optimality scales as $$\tilde{\mathcal{O}}(1/\epsilon \cdot \text{poly}(T))$$ for a given horizon (Chen et al., 25 Sep 2024).
• Low-rank representations and scalability: Non-stationary value functions in high-dimensional finite-horizon MDPs are structured as high-order tensors, and low-rank decompositions enable efficient parameterization and tractable learning, with block-coordinate descent or TD-style algorithms yielding convergence even when system dynamics are unknown (Rozada et al., 17 Jan 2025). This is particularly valuable in large-scale resource allocation and scheduling.

7. Extensions and Future Directions

Contemporary research leverages the finite-horizon framework for quantum speedup, non-Markovian stochastic control, and hierarchical real-time planning:

• Quantum algorithms: Finite-horizon optimal control (e.g., LQG) is reformulated for quantum hardware using block encodings and Quantum Singular Value Transformation (QSVT), yielding asymptotic polylogarithmic complexity in state dimension $n$ and linear scaling in $T$, assuming matrices are well-conditioned and block-encodable (Dehaghani et al., 14 Jul 2025).
• Robust non-Markovian impulse control: Systems described by functional SDEs on finite horizons are addressed via interconnected, obliquely reflected backward SDEs, with existence and uniqueness accessibly proved using Picard iteration (Perninge, 2021).
• Hierarchical factorization in MAPF: Receding-horizon, parallelizable algorithms for large-scale path planning in dynamic environments exploit finite-horizon groupwise planning and online conflict resolution, achieving empirically substantial reductions in time-to-first-action while maintaining solution quality (Li et al., 12 May 2025).

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In sum, the finite-horizon framework is a foundational construct in modern dynamic decision-making. Its explicit incorporation of planning length fundamentally impacts computational requirements, structural properties of solutions, theoretical guarantees, and practical performance across a diverse set of domains—from robust control and planning to the game-theoretic modeling of interacting agents, from learning in high-dimensional reinforcement learning to model selection in systems identification. Whether as approximation, regularization, computational necessity, or explicit feature of the real-world environment, finite-horizon modeling provides precise and scalable machinery for rigorously controlling complex systems over time.