Generic controllability of equivariant systems and applications to particle systems and neural networks

There exist many examples of systems which have some symmetries, and which one may monitor with symmetry-preserving controls. Since symmetries are preserved along the evolution, full controllability is not possible, and controllability has to be considered inside sets of states with the same symmetries. We prove that generic systems with symmetries are controllable in this sense. This result has several applications, for instance, (i) generic controllability of particle systems when the kernel of interaction between particles plays the role of a mean-field control; (ii) generic controllability for families of vector fields on manifolds with boundary; (iii) universal interpolation for neural networks architectures with “generic” self-attention-type layers – a type of layer ubiquitous in recent neural network architectures, e.g., in the Transformers architecture. The tools we develop could help address various other questions of control of equivariant systems.