# Local Linearity of LLMs Enables Activation Steering via Model-Based Linear Optimal Control

arXiv: https://arxiv.org/abs/2604.19018 · HTML: https://mlsanity.com/paper/2604.19018/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in LLM preference optimization methods, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 53 of its 65 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Placed in

- LLM alignment / RLHF / preference optimisation / [LLM preference optimization methods](https://mlsanity.com/alignment/llm-preference-optimization-methods/)
- Mechanistic interpretability / [Steering and probing for LLM safety](https://mlsanity.com/mechinterp/steering-and-probing-for-llm-safety/)

## Builds on (most cited first, 30 of 53)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Jailbroken: How Does LLM Safety Training Fail?](https://mlsanity.com/paper/2307.02483/) (2023)
- [HarmBench: A Standardized Evaluation Framework for Automated Red Teaming and Robust Refusal](https://mlsanity.com/paper/2402.04249/) (2024)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [RealToxicityPrompts: Evaluating Neural Toxic Degeneration in Language Models](https://mlsanity.com/paper/2009.11462/) (2020)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [A General Language Assistant as a Laboratory for Alignment](https://mlsanity.com/paper/2112.00861/) (2021)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Plug and Play Language Models: A Simple Approach to Controlled Text Generation](https://mlsanity.com/paper/1912.02164/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets](https://mlsanity.com/paper/2310.06824/) (2023)
- [RRHF: Rank Responses to Align Language Models with Human Feedback without tears](https://mlsanity.com/paper/2304.05302/) (2023)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Trustworthy LLMs: a Survey and Guideline for Evaluating Large Language Models' Alignment](https://mlsanity.com/paper/2308.05374/) (2023)
- [Contrastive Preference Optimization: Pushing the Boundaries of LLM Performance in Machine Translation](https://mlsanity.com/paper/2401.08417/) (2024)
- [Preference Ranking Optimization for Human Alignment](https://mlsanity.com/paper/2306.17492/) (2023)
- [LLMs Know More Than They Show: On the Intrinsic Representation of LLM Hallucinations](https://mlsanity.com/paper/2410.02707/) (2024)
- [Programming Refusal with Conditional Activation Steering](https://mlsanity.com/paper/2409.05907/) (2024)
- [ReFT: Representation Finetuning for Language Models](https://mlsanity.com/paper/2404.03592/) (2024)
- [Defending Large Language Models Against Jailbreaking Attacks Through Goal Prioritization](https://mlsanity.com/paper/2311.09096/) (2023)
- [Analyzing the Generalization and Reliability of Steering Vectors](https://mlsanity.com/paper/2407.12404/) (2024)

## Built on by (most cited first, 9 of 9)

- [Activation Steering of Video Generation Models via Reduced-Order Linear Optimal Control](https://mlsanity.com/paper/2606.04775/) (2026)
- [Steering Robustness into World Action Models via Mechanistic Interpretability and Optimal Control](https://mlsanity.com/paper/2607.14943/) (2026)
- [ObserverBench: Testing Mechanistic Estimates for Intervention and Control](https://mlsanity.com/paper/2609.03026/) (2026)
- [Activation Steering Transfer to Agents: One Gain Ratio Does Not Identify Potency and Efficacy](https://mlsanity.com/paper/2607.09156/) (2026)
- [Activation-Space Order-Swap Geometry: A Site-Asymmetry Audit](https://mlsanity.com/paper/2608.25315/) (2026)
- [Metacognitive Steering: Learning the Structure of Scientific Judgment](https://mlsanity.com/paper/2609.16245/) (2026)
- [Replay-Gated Neural Execution: Decoupling Persistent Behavioral Specifications from Neural Realizations in Frozen Language Models](https://mlsanity.com/paper/2609.22243/) (2026)
- [Adaptive Multi-Value Control in LLMs via Causal Activation Steering](https://mlsanity.com/paper/2609.30405/) (2026)
- [From Latent Space to Jacobian Space: Measuring, Evading, and Training Against Safety-Content Accessibility](https://mlsanity.com/paper/2610.04316/) (2026)

## Abstract (arXiv)

Inference-time LLM alignment methods, particularly activation steering, offer an alternative to fine-tuning by directly modifying activations during generation. Existing methods, however, often rely on non-anticipative interventions that ignore how perturbations propagate through transformer layers and lack online error feedback, resulting in suboptimal, open-loop control. To address this, we show empirically that, despite the nonlinear structure of transformer blocks, layer-wise dynamics across multiple LLM architectures and scales are well-approximated by locally-linear models. Exploiting this property, we model LLM inference as a linear time-varying dynamical system and adapt the classical linear quadratic regulator to compute feedback controllers using layer-wise Jacobians, steering activations toward desired semantic setpoints in closed-loop with minimal computational overhead and no offline training. We also derive theoretical bounds on setpoint tracking error, enabling formal guarantees on steering performance. Using a novel adaptive semantic feature setpoint signal, our method yields robust, fine-grained behavior control across models, scales, and tasks, including state-of-the-art modulation of toxicity, truthfulness, refusal, and arbitrary concepts, surpassing baseline steering methods. Our code is available at: https://github.com/trustworthyrobotics/lqr-activation-steering