Recently, I have been working on fundamental linear control problems—such as LQR, H∞ control, and LQ differential games—using tools from nonconvex optimization, nonsmooth analysis, and duality theory. My work includes deriving new solution methods, analyzing convergence rates that were previously unclear, and establishing strong duality results. I also worked on linear system control when I was in Japan.
These classical problems are often considered “solved” by 2000s. Because of that, I am sometimes asked why I am still working on them. This is a fair question! Of course, I do not mean I will remain in this kind of research forever.
Still, I do believe there are reasonable motivations behind it, and I am now seeing many fruitful results (though, to be honest, I also simply enjoy control theory and optimization).
Here are some of the motivations and potential benefits.
1. Important properties are still not fully understood
Even in areas that appear mature, there are often gaps in our understanding. In fact, relatively recent work has revealed striking properties of LQR, such as gradient dominance, showing that the theory still had a room for further investigations.
More recently, we have been studying unresolved issues in robust controller design algorithms—such as those used in practice and implemented in tools like MATLAB (e.g., convergence guarantees and rates), with a cool application:
https://www.math.univ-toulouse.fr/~noll/PAPERS/rosetta.pdf
By applying modern nonconvex optimization theory, we clarified some aspects that remained open for decades, such as the sublinear convergence rate and weak convexity. We are now working on the further algorithmic improvement and extensions.
Policy Optimization in Robust Control: Weak Convexity and...
A deeper theoretical understanding may eventually lead to new applications. In addition, many recent optimization results, such as global optimization of nonconvex problems (e.g., nonconvexs QCQPs), are also of huge interest by themselves and would deserve further investigations in the control literature!
2. Providing an alternative pathway to understand modern control
Modern control theory is well established, with many textbooks available. However, learning it typically requires a specific background (classical control, differential equations, etc.).
If we can reconstruct key results from a different perspective—for example, through convex optimization, semidefinite programming, and duality—this creates an alternative entry point. It is in fact possible in many cases, while many of them are often overlooked or scattered in the literature in an incomplete form. In a time when the amount of knowledge keeps growing, having multiple ways to access the same theory is valuable.
We clarified analytical solutions to LQR, H∞ control, and LQ game problems from an optimization and duality viewpoint. Since optimization is central to modern approaches like MPC, bridging the gap between classical control and optimization could offer a new way to learn the subject. This may also attract more people in optimization, ML, and signal processing to control.
Revisiting Strong Duality, Hidden Convexity, and Gradient...