Learning for control of dynamical systems with formal guarantees remains a challenging task. This paper proposes a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing nonlinear controller. The SMT solver then verifies that the candidate Lyapunov function indeed satisfies the Lyapunov conditions. We provide theoretical guarantees of the proposed learning framework in terms of the closed-loop stability for the unknown nonlinear system. We illustrate the effectiveness of the approach with a set of numerical experiments.Preprint. Under review.

The center of mass of an operator A (denoted St(A), and called in this paper as the Stampfli point of A) was introduced by Stampfli in his Pacific J. Math (1970) paper as the unique λ ∈ C delivering the minimum value of A − λI . We derive some results concerning the location of St(A) for several classes of operators, including 2-by-2 block operator matrices with scalar diagonal blocks and 3-by-3 matrices with repeated eigenvalues. We also show that for almost normal A its Stampfli point lies in the convex hull of the spectrum, which is not the case in general. Some relations between the property St(A) = 0 and Roberts orthogonality of A to the identity operator are established.

The center of mass of an operator A (denoted St(A) , and called in this paper as the Stampfli point of A ) was introduced by Stampfli in his Pacific J. Math (1970) paper as the unique λ ∈ C delivering the minimum value of A − λ I . We derive some results concerning the location of St(A) for several classes of operators, including 2-by-2 block operator matrices with scalar diagonal blocks and 3-by-3 matrices with repeated eigenvalues. We also show that for almost normal A its Stampfli point lies in the convex hull of the spectrum, which is not the case in general. Some relations between the property St(A) = 0 and Roberts orthogonality of A to the identity operator are established.