Conical Lie Groups in Control Theory

From AlphaGo's unrivaled chess prowess to ChatGPT’s uncanny 
mastery of language, Reinforcement Learning (RL) has begun to 
revolutionize the world as we know it. However, such RL successes are 
lacking for control of continuous-time (CT) dynamical systems like cars 
and airplanes. Recent studies show that CT-RL control problems face 
unique numerical challenges, particularly in regards to conditioning 
issues associated with the underlying linear regressions used to solve 
for the neural network (NN) weights. In this seminar, I will discuss a 
new method of improving RL problem conditioning via nonsingular linear 
transformations of the system state data. The relationship between the 
control solution and the transformation has a particular structure of a 
Lie group homomorphism which is homogeneous with respect to nonzero 
scaling of the transformation matrix. This motivates a concept of a 
"conical" Lie group (with Lie group structure plus a particular 
"scaling" smooth action) and homomorphisms between such Lie groups. 
These objects show great potential in enabling RL to address real-world 
CT control problems.