# LMI Properties and Applications in Systems, Stability, and Control Theory

arXiv: https://arxiv.org/abs/1903.08599 · HTML: https://mlsanity.com/paper/1903.08599/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2019.

## Built on by (most cited first, 11 of 11)

- [Safe Adaptive Learning-based Control for Constrained Linear Quadratic Regulators with Regret Guarantees](https://mlsanity.com/paper/2111.00411/) (2021)
- [On Imitation Learning of Linear Control Policies: Enforcing Stability and Robustness Constraints via LMI Conditions](https://mlsanity.com/paper/2103.12945/) (2021)
- [Pick-to-Learn for Systems and Control: Data-driven Synthesis with State-of-the-art Safety Guarantees](https://mlsanity.com/paper/2512.04781/) (2025)
- [Linear Matrix Inequality Approaches to Koopman Operator Approximation](https://mlsanity.com/paper/2102.03613/) (2021)
- [System Norm Regularization Methods for Koopman Operator Approximation](https://mlsanity.com/paper/2110.09658/) (2021)
- [Closed-Loop Koopman Operator Approximation](https://mlsanity.com/paper/2303.15318/) (2023)
- [Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator](https://mlsanity.com/paper/2410.01057/) (2024)
- [L2RU: a Structured State Space Model with prescribed L2-bound](https://mlsanity.com/paper/2503.23818/) (2025)
- [Operator-Theoretic Foundations and Policy Gradient Methods for General MDPs with Unbounded Costs](https://mlsanity.com/paper/2603.17875/) (2026)
- [Dissipative Imitation Learning for Discrete Dynamic Output Feedback Control with Sparse Data Sets](https://mlsanity.com/paper/2309.06658/) (2023)
- [Imitation Learning from Observations: An Autoregressive Mixture of Experts Approach](https://mlsanity.com/paper/2411.08232/) (2024)

## Abstract (arXiv)

Linear matrix inequalities (LMIs) commonly appear in systems, stability, and control applications. Many analysis and synthesis problems in these areas can be solved as feasibility or optimization problems subject to LMI constraints. Although most well-known LMI properties and manipulation tricks, such as the Schur complement and the congruence transformation, can be found in standard references, many useful LMI properties are scattered throughout the literature. The purpose of this document is to collect and organize properties, tricks, and applications related to LMIs from a number of references together in a single document. In this sense, the document can be thought of as an "LMI encyclopedia" or "LMI cookbook." Proofs of the properties presented in this document are not included when they can be found in the cited references in the interest of brevity. Illustrative examples are included whenever necessary to fully explain a certain property. Multiple equivalent forms of LMIs are often presented to give the reader a choice of which form may be best suited for a particular problem at hand. The equivalency of some of the LMIs in this document may be straightforward to more experienced readers, but the authors believe that some readers may benefit from the presentation of multiple equivalent LMIs.