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arxiv: 2604.21577 · v1 · submitted 2026-04-23 · 🧮 math.OC

Infinite Horizon Optimal Control Problems with Discount Factors

Eduardo Casas , Karl Kunisch This is my paper

Pith reviewed 2026-05-09 21:27 UTC · model grok-4.3

classification 🧮 math.OC
keywords optimal controlinfinite horizonsemilinear parabolic equationsdiscount factorsoptimality conditionsfinite horizon approximationconvergence
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The pith

Allowing different discount factors for the cost and state enables first- and second-order optimality conditions for infinite-horizon control of semilinear parabolic equations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives optimality conditions for problems that minimize a discounted cost over an infinite time horizon, subject to a semilinear parabolic state equation and box constraints on the control. It permits the discount factor multiplying the running cost integral to be chosen independently of the discount factor that multiplies the state tracking term. This independence is shown to be crucial for the second-order sufficient conditions to hold for the nonlinearities considered in the state equation. The analysis concludes by proving that the optimal values and controls of the infinite-horizon problem are the limits of those from finite-horizon truncations, together with a rate of convergence.

Core claim

The paper derives first-order optimality conditions using the adjoint equation and a variational inequality. For second-order conditions, it shows that the quadratic form in the second variation must account for the differing discounts to ensure positivity under the given nonlinearities. Additionally, the infinite-horizon problem is shown to be the limit of finite-horizon problems as the horizon tends to infinity, with a proven rate of convergence for the optimal values and controls.

What carries the argument

The objective functional with independently chosen exponential discount factors for the running cost and the state penalty term.

Load-bearing premise

The nonlinearities in the semilinear parabolic state equation are of a form that requires the discount factor on the state to differ from the one on the cost in order for second-order sufficient conditions to hold.

What would settle it

A concrete example of the semilinear parabolic equation and cost where equal discount factors are used and the second variation of the cost at a first-order stationary point is found to be negative.

read the original abstract

This paper is dedicated to the analysis of infinite horizon optimal control problems subject to semilinear parabolic equations with constraints on the controls and discounted cost functionals. The discount factors on the cost and the state components are allowed to differ from each other. First-order as well as second-order optimality conditions are derived and the importance of allowing different discount factors for the second-order analysis for the class of nonlinearities under consideration is demonstrated. Finally convergence and rate of convergence for the approximation of the infinite horizon problem by a family of finite horizon problems is proven.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript analyzes infinite-horizon optimal control problems governed by semilinear parabolic PDEs subject to control constraints and discounted cost functionals, where distinct discount factors are permitted for the cost and state components. It derives first-order and second-order optimality conditions, demonstrates that equal discount factors are insufficient for valid second-order conditions under the considered class of nonlinearities, and proves convergence together with convergence rates for the approximation of the infinite-horizon problem by a family of finite-horizon truncations.

Significance. If the derivations hold, the work supplies a technically sound extension of optimality theory for infinite-horizon parabolic control problems. The explicit allowance for unequal discount factors addresses a concrete obstruction in the second variation for the given nonlinearities, while the convergence result with rates furnishes a rigorous justification for numerical truncation. These elements strengthen the applicability of infinite-horizon formulations in PDE-constrained optimization.

minor comments (2)
  1. [§3] The statement of the admissible control set and the precise growth conditions on the nonlinearity (presumably in §2 or §3) should be cross-referenced explicitly when the second-order remainder term is estimated, to make the necessity of distinct discount factors fully transparent without backtracking.
  2. [§5] In the convergence proof, the dependence of the rate constant on the discount factors should be stated explicitly (e.g., in the final estimate of Theorem 5.3 or its analogue) rather than left implicit in the stability constants.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment, including the recommendation for minor revision. The summary accurately captures the main contributions regarding first- and second-order optimality conditions for infinite-horizon problems with possibly distinct discount factors, as well as the convergence analysis for finite-horizon approximations.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from standard first- and second-order optimality conditions for semilinear parabolic control problems, with the allowance for distinct discount factors introduced explicitly as a modeling choice to control quadratic remainders in the second variation. The convergence result for finite-horizon approximations follows from stability estimates once the optimality conditions are established. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard well-posedness assumptions for semilinear parabolic PDEs and discounted infinite-horizon integrals; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The semilinear parabolic state equation admits unique solutions in appropriate function spaces under the given control constraints.
    Required to define the admissible control-to-state map for the optimization problem.
  • domain assumption The discount factors are positive real numbers so that the infinite-horizon cost integrals converge.
    Necessary for the problem to be well-posed.

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Reference graph

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