Ibon Gracia

Chun-Wei Kong, Sebastian Escobar, Ibon Gracia et al.

Due to their expressive power, neural networks (NNs) are promising templates for functional optimization problems, particularly for reach-avoid certificate generation for systems governed by stochastic differential equations (SDEs). However, ensuring hard-constraint satisfaction remains a major challenge. In this work, we propose two constraint-driven training frameworks with guarantees for supermartingale-based neural certificate construction and controller synthesis for SDEs. The first approach enforces certificate inequalities via domain discretization and a bound-based loss, guaranteeing global validity once the loss reaches zero. We show that this method also enables joint NN controller-certificate synthesis with hard guarantees. For high-dimensional systems where discretization becomes prohibitive, we introduce a partition-free, scenario-based training method that provides arbitrarily tight PAC guarantees for certificate constraint satisfaction. Benchmarks demonstrate scalability of the bound-based method up to 5D, outperforming the state of the art, and scalability of the scenario-based approach to at least 10D with high-confidence guarantees.

Ibon Gracia, Qi Heng Ho, Luca Laurenti et al.

We present a provably safe sampling-based motion planning algorithm for robotic systems affected by random disturbances of unknown distribution. We consider systems with linear or linearizable dynamics evolving in workspace with arbitrary-shaped obstacles subject to state and control constraints. Safety requirements are formulated as chance-constraints. Our approach leverages data from trajectories of the system to learn a Wasserstein ambiguity tube, i.e., a sequence of ambiguity sets, which contains the trajectory of the system's state distribution with high confidence. This ambiguity tube is then used in a probabilistically complete algorithm to grow a sampling-based motion planning tree that respects the constraints of the problem. We show that learning several lower-dimensional ambiguity tubes instead of a single high-dimensional one effectively reduces the conservatism and boosts scalability. Additionally, we design an efficient bandit-based validity checker that remarkably increases the empirical performance of our approach without sacrificing probabilistic completeness. Case studies show our algorithm finds valid plans in cluttered environments under strict safety thresholds, outperforming state-of-the-art methods.

Ibon Gracia, Morteza Lahijanian

We study the asymptotic optimality of abstraction-based control synthesis algorithms. Specifically, we consider uncertain MDP (UMDP) abstraction, and investigate whether refinement leads to optimal results, i.e., an optimal controller and zero error bound. Additionally, we study completeness of abstraction-refinement algorithms, i.e., that the algorithm produces near-optimal results in finite time. The focus is on nonlinear stochastic systems with general vector fields and temporal logic specifications. We present an algorithm that abstracts the system into a UMDP and synthesizes a controller with performance guarantees via robust dynamic programming. Then, the algorithm iteratively refines the abstraction until a near-optimality criterion is met. A thorough theoretical analysis reveals a sufficient condition, which we denote vanishing ambiguity, guaranteeing asymptotic optimality of the abstraction process and completeness of the algorithm. We show that set-valued MDP abstractions satisfy this criterion, whereas interval MDP abstractions lack such a guarantee.