Controllability of random systems: Universality and minimal controllability

For a large class of random matrices $A$ and vectors $b$, we show that linear systems formed from the pair $(A,b)$ are controllable with high probability. Despite the fact that minimal controllability problems are, in general, NP-hard, we establish universality results for the minimal controllability of random systems. Our proof relies on the recent developments of Nguyen-Tao-Vu concerning gaps between eigenvalues of random matrices.