1. E.D. Sontag. Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning. Systems and Control Letters, 161:105138, 2022. Note: Important: there is an error in the paper. For the LQR application, the paper only shows iISS, not ISS. See the paper Small-disturbance input-to-state stability of perturbed gradient flows: Applications to LQR problem for details.[PDF] Keyword(s): iss, input to state stability, data-driven control, gradient systems, steepest descent, model-free control. Abstract:

   Recent work on data-driven control and reinforcement learning has renewed interest in a relatively old field in control theory: model-free optimal control approaches which work directly with a cost function and do not rely upon perfect knowledge of a system model. Instead, an "oracle" returns an estimate of the cost associated to, for example, a proposed linear feedback law to solve a linear-quadratic regulator problem. This estimate, and an estimate of the gradient of the cost, might be obtained by performing experiments on the physical system being controlled. This motivates in turn the analysis of steepest descent algorithms and their associated gradient differential equations. This paper studies the effect of errors in the estimation of the gradient, framed in the language of input to state stability, where the input represents a perturbation from the true gradient. Since one needs to study systems evolving on proper open subsets of Euclidean space, a self-contained review of input to state stability definitions and theorems for systems that evolve on such sets is included. The results are then applied to the study of noisy gradient systems, as well as the associated steepest descent algorithms.

1. A.C.B de Olivera, L. Cui, and E. D. Sontag. Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems. In Proc. 64th IEEE Conference on Decision and Control (CDC), 2025. Note: Submitted. Keyword(s): gradient dominance, gradient flows, LQR, reinforcement learning. Abstract:

   This work explores generalizations of the Polyak-Lojasiewicz inequality (PLI) and their implications for the convergence behavior of gradient flows in optimization problems. Motivated by the continuous-time linear quadratic regulator (CT-LQR) policy optimization problem -- where only a weaker version of the PLI is characterized in the literature -- this work shows that while weaker conditions are sufficient for global convergence to, and optimality of the set of critical points of the cost function, the "profile" of the gradient flow solution can change significantly depending on which "flavor" of inequality the cost satisfies. After a general theoretical analysis, we focus on fitting the CT-LQR policy optimization problem to the proposed framework, showing that, in fact, it can never satisfy a PLI in its strongest form. We follow up our analysis with a brief discussion on the difference between continuous- and discrete-time LQR policy optimization, and end the paper with some intuition on the extension of this framework to optimization problems with L1 regularization and solved through proximal gradient flows.




2. M. Sznaier, A. Olshevsky, and E.D. Sontag. The role of systems theory in control oriented learning. In Proc. 25th Int. Symp. Mathematical Theory of Networks and Systems (MTNS 2022), 2022. Note: To appear.[PDF] Keyword(s): control oriented learning, neural networks, reinforcement learning, feedback control, machine learning. Abstract:

   Systems theory can play an important in unveiling fundamental limitations of learning algorithms and architectures when used to control a dynamical system, and in suggesting strategies for overcoming these limitations. As an example, a feedforward neural network cannot stabilize a double integrator using output feedback. Similarly, a recurrent NN with differentiable activation functions that stabilizes a non-strongly stabilizable system must be itself open loop unstable, a fact that has profound implications for training with noisy, finite data. A potential solution to this problem, motivated by results on stabilization with periodic control, is the use of neural nets with periodic resets, showing that indeed systems theoretic analysis is instrumental in developing architectures capable of controlling certain classes of unstable systems. This short conference paper also argues that when the goal is to learn control oriented models, the loss function should reflect closed loop, rather than open loop model performance, a fact that can be accomplished by using gap-metric motivated loss functions.

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