Small-disturbance input-to-state stability of perturbed gradient flows: Applications to LQR problem

Leilei Cui, Zhong Ping Jiang, Eduardo D. Sontag

Research output: Contribution to journalArticlepeer-review


This paper studies the effect of perturbations on the gradient flow of a general nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectories of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. We show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow.

Original languageEnglish (US)
Article number105804
JournalSystems and Control Letters
StatePublished - Jun 2024


  • Gradient systems
  • Input-to-state stability (ISS)
  • Learning-based control
  • Linear quadratic regulator (LQR)
  • Policy optimization

ASJC Scopus subject areas

  • Control and Systems Engineering
  • General Computer Science
  • Mechanical Engineering
  • Electrical and Electronic Engineering


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