Learning-based methods have achieved remarkable performance in robotic control, yet providing formal guarantees for learned controllers remains an open challenge. The central question motivating this work is how control-theoretic structure can serve as an inductive bias for learning, producing controllers that are both flexible and certifiable. The key insight is that global properties such as stability and safety reduce to pointwise conditions that neural networks can learn to satisfy. A Lyapunov function certifies stability through a condition that must hold at every state in a region; a barrier function certifies safety through a condition that must hold throughout the safe set. Rather than optimizing over expensive trajectory rollouts, one can sample states and penalize violations of these pointwise conditions. If the conditions are satisfied throughout the relevant region of state space, the global guarantee follows. Three complementary strategies are developed for integrating control structure with learning: training on pointwise certificate conditions, learning inputs to structured controllers that provide guarantees conditional on quantities like uncertainty estimates, and enforcing structure architecturally so that desired properties hold by construction. These strategies are validated on neural ordinary differential equations for certified stability and safety, stereo vision systems that learn perception uncertainty for safe quadruped navigation, and Koopman operator learning with guaranteed linear latent dynamics for predictive control of floating platforms. The primary application domain is underactuated legged locomotion. Zero dynamics, the residual dynamics that remain when controlled outputs have been driven to zero, provide a framework for coordinating actuated degrees of freedom with passive dynamics. A zero dynamics policy is a learned mapping from unactuated to actuated states that defines a stable invariant manifold. Theoretical results establish existence of such policies for locally controllable systems and prove that manifold stability implies full-state stability. Hardware experiments on a bipedal walker and a hopping robot validate these results: the hopper achieves over three thousand hops across stairs, ramps, and narrow bridges with robust disturbance rejection. These experiments demonstrate that control-theoretic structure and neural network learning can be combined to achieve both formal guarantees and robust physical performance.
Ivan Dario Jimenez Rodriguez (Tue,) studied this question.