Beyond Test-Time Memory: State-Space Optimal Control for LLM Reasoning

Abstract

Associative memory has long underpinned the design of sequential models. Beyond recall, humans reason by *projecting future states and selecting goal-directed actions*, a capability that modern language models increasingly require but do not natively encode. While prior work uses reinforcement learning or test-time training, planning remains external to the model architecture. We formulate reasoning as *optimal control* and introduce the *Test-Time Control (TTC)* layer, which performs finite-horizon *LQR* planning over latent states at inference time, represents a *value function* within neural architectures, and leverages it as the nested objective to enable *planning before prediction*. To ensure scalability, we derive a hardware-efficient LQR solver based on a symplectic formulation and implement it as a fused CUDA kernel, enabling parallel execution with minimal overhead. Integrated as an adapter into pretrained LLMs, TTC layers improve mathematical reasoning performance by up to +27.8% on MATH-500 and 2-3$\times$ Pass@8 improvements on AMC and AIME, demonstrating that embedding optimal control as an architectural component provides an effective and scalable mechanism for reasoning beyond test-time training.