Estimate the distance of closest approach between points along trajectories of x'=f(t, x) starting from an initial set X0 and an unsafe set Xu over a time horizon of [0, T]. If possible, recover the trajectories that attain the closest approach.

An occupation-measure framework is used to find a convergeng sequence of upper bounds in rising degree to the true minimal distance value. The approximate-optimal trajectories may be recovered if moment matrices satisfy rank conditions (up to numerical accuracy).

• Gloptipoly3: https://homepages.laas.fr/henrion/software/gloptipoly/. A customized version is included in as a submodule to this repository (gloptipoly_mod)
• YALMIP: https://yalmip.github.io/
• Mosek: https://www.mosek.com/ (or any solver compatible with YALMIP)

All code is written and tested on Matlab R2021a.

The distance_manager class has two input arguments: lsupp and f. The object lsupp of type unsafe_support is the options structure that defines properties of the system, such as its variables, the distance function, and the support set of (X, X0, Xu). The input f are the dynamics of the system.

Execute distance_manager.run(order) to formulate and solve the distance estimation program. Call distance_manager.loc.recover() to attempt rank-based recovery of near-optimal trajectories. The orderis the relaxation order involving moments of degree2*order`.

Examine and run experiments/flow_distance_test.m as an example. The README in the experiments folder describes all test scripts.

https://arxiv.org/abs/2110.14047 "Bounding the Distance to Unsafe Sets with Convex Optimization"

https://homepages.laas.fr/vmagron/brainpop/DistanceJared.pdf Presentation given at BrainPOP seminar at LAAS-CNRS on June 28, 2021

For comments and questions please email Jared Miller.