Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.27579

A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case

Hongwei Mei
View original source ↗Read source document ↗Export metadata ↓

Abstract

The existing theory of stochastic linear-quadratic (LQ) optimal control problems focuses on open-loop and closed-loop solvability. For finite horizon problems, it is widely accepted that those two types of solvability are not equivalent in general, as demonstrated by explicit counterexamples. In contrast, they coincide in standard infinite-horizon formulations with constant coefficients under suitable stabilizability and integrability assumptions. This paper aims at challenging this well-established paradigm by developing a novel framework of closed-loop solvability, namely generalized closed-loop solvability, which removes the $L^2$-integrability constraint on the feedback strategies from the existing formulation. It will be proven that open-loop and generalized closed-loop solvability are equivalent and the classical closed-loop solvablity is a particular case only when the admissible feedback strategy is $L^2$-integrable over the time horizon. In other words, the previously established non-equivalence between open-loop and closed-loop solvability stems solely from the limitations of the classical formulation which requires the $L^2$-integrability of the admissible feedback strategies over the time-horizon. Therefore, this paper unifies the theory of the solvabilitiy for stochastic LQ control problems. Moreover, our results offer a novel insight to stochastic LQ control problems: open-loop/generalized closed-loop solvability is essentially equivalent to piecewisely local closed-loop solvability in the classical sense. The current paper is focused on homogeneous cases only and non-homogeneous cases will be studied in a subsequent paper.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hongwei Mei. 2026-10-20. A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case. https://arxiv.org/abs/2610.27579

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Exploring chordal sparsity in semidefinite programming with sparse plus low-rank data matrices

Semidefinite programming (SDP) problems are challenging to solve because of their high dimensionality. However, solving sparse SDP problems with small tree-width are known to be relatively easier because: (1) they can be decomposed into smaller multi-block SDP problems through chordal conversion; (2) they have low-rank optimal solutions. In this paper, we study more general SDP problems whose coefficient matrices have sparse plus low-rank (SPLR) structure. We develop a unified framework to convert such problems into sparse SDP problems with bounded tree-width. Based on this, we derive rank bounds for SDP problems with SPLR structure, which are tight in the worst case.

math.OC↗
arXiv2026

Bellman Optimality of Average-Reward Robust Markov Decision Processes with a Constant Gain

Learning and optimal control under robust Markov decision processes (MDPs) have received increasing attention, yet most existing theory, algorithms, and applications focus on finite-horizon or discounted models. Long-run average-reward formulations, while natural in many operations research and management contexts, remain underexplored. This is primarily because the dynamic programming foundations are technically challenging and only partially understood, with several fundamental questions remaining open. This paper steps toward a general framework for average-reward robust MDPs by analyzing the constant-gain setting. We study the average-reward robust control problem with possible information asymmetries between the controller and an S-rectangular adversary. Our analysis centers on the constant-gain robust Bellman equation, examining both the existence of solutions and their relationship to the optimal average reward. Specifically, we identify when solutions to the robust Bellman equation characterize the optimal average reward and stationary policies, and we provide one-sided weak communication conditions ensuring solutions' existence. These findings expand the dynamic programming theory for average-reward robust MDPs and lay a foundation for robust dynamic decision making under long-run average criteria in operational environments.

math.OC↗
arXiv2026

Service-Induced Congestion in Memory-Constrained LLM Serving

In large language model (LLM) serving, each request accumulates persistent graphics processing unit (GPU) memory during service as its key-value cache grows with every generated token. Under high concurrency, aggregate memory usage therefore increases endogenously over time: the service process itself creates future capacity pressure. When memory capacity is exceeded, systems evict active requests, discarding cached state and restarting them later, which wastes computation and reduces throughput. We develop a discrete-time dynamical model of memory-constrained LLM inference that captures admission, memory growth, and eviction under continuous batching. In the saturated-input regime, the system admits both eviction-free fixed points and limit cycles with evictions. For homogeneous workloads, we show that the eviction-free equilibrium is unstable and that, except for a Lebesgue-measure-zero exact-capture set, the system converges to a unique worst-case limit cycle that is asymptotically stable outside this exceptional set, with throughput losses as large as 50%. For heterogeneous workloads, we prove a stability criterion in the two-class common-input setting and explain how the survival-polynomial mechanism generalizes to multiple classes and heterogeneous-input lengths. Under an input-dominated scaling regime, coprime decoding lengths stabilize the eviction-free equilibrium, while non-coprime lengths create synchronized modes that drive instability. These results characterize when workload heterogeneity desynchronizes completions and helps stabilize memory-constrained serving. More broadly, we identify service-induced congestion as a structural instability mechanism and derive scheduling design principles for sustaining high throughput.

math.OC↗