In this paper, we describe an abstraction based method for synthesizing a state based switching control for stabilizing a family of dynamical systems. Given a set of dynamical systems and a set of polyhedral switching surfaces, the algorithm synthesizes a strategy that assigns to every surface the linear dynamics to switch to at the surface. Our algorithm constructs a finite game graph that consists of the switching surfaces as the existential nodes and the choices of the dynamics as the universal nodes. In addition, the edges capture quantitative information about the evolution of the distance of the state from the equilibrium point along the executions. A switching strategy for the family of dynamical systems is extracted by finding a strategy on the game graph which results in plays having a bounded weight. Such a strategy is obtained by reducing the problem to the strategy synthesis for an energy game, which is a well-studied problem in the literature. We have implemented our algorithm for polyhedral inclusion dynamics and linear dynamics. We illustrate our algorithm on examples from these two classes of systems.