We propose a novel framework for learning stabilizable nonlinear dynamical\nsystems for continuous control tasks in robotics. The key contribution is a\ncontrol-theoretic regularizer for dynamics fitting rooted in the notion of\nstabilizability, a constraint which guarantees the existence of robust tracking\ncontrollers for arbitrary open-loop trajectories generated with the learned\nsystem. Leveraging tools from contraction theory and statistical learning in\nReproducing Kernel Hilbert Spaces, we formulate stabilizable dynamics learning\nas a functional optimization with convex objective and bi-convex functional\nconstraints. Under a mild structural assumption and relaxation of the\nfunctional constraints to sampling-based constraints, we derive the optimal\nsolution with a modified Representer theorem. Finally, we utilize random matrix\nfeature approximations to reduce the dimensionality of the search parameters\nand formulate an iterative convex optimization algorithm that jointly fits the\ndynamics functions and searches for a certificate of stabilizability. We\nvalidate the proposed algorithm in simulation for a planar quadrotor, and on a\nquadrotor hardware testbed emulating planar dynamics. We verify, both in\nsimulation and on hardware, significantly improved trajectory generation and\ntracking performance with the control-theoretic regularized model over models\nlearned using traditional regression techniques, especially when learning from\nsmall supervised datasets. The results support the conjecture that the use of\nstabilizability constraints as a form of regularization can help prune the\nhypothesis space in a manner that is tailored to the downstream task of\ntrajectory generation and feedback control, resulting in models that are not\nonly dramatically better conditioned, but also data efficient.\n