We have often observed unexpected state transitions of complex systems. We are thus interested in how to steer a complex system from an unexpected state to a desired state. Here we introduce the concept of transittability of complex networks, and derive a new sufficient and necessary condition for state transittability which can be efficiently verified. We define the steering kernel as a minimal set of steering nodes to which control signals must directly be applied for transition between two specific states of a network, and propose a graph-theoretic algorithm to identify the steering kernel of a network for transition between two specific states. We applied our algorithm to 27 real complex networks, finding that sizes of steering kernels required for transittability are much less than those for complete controllability. Furthermore, applications to regulatory biomolecular networks not only validated our method but also identified the steering kernel for their phenotype transitions.