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The topology transition problem of transmission networks is becoming increasingly crucial with topological flexibility more widely leveraged to promote high renewable penetration. This paper proposes a novel methodology to address this problem. Aiming at achieving a bumpless topology transition regarding both static and dynamic performance, this methodology utilizes various eligible control resources in transmission networks to cooperate with the optimization of line-switching sequence. Mathematically, a composite formulation is developed to efficiently yield bumpless transition schemes with AC feasibility and stability both ensured. With linearization of all non-convexities involved and tractable bumpiness metrics, a convex mixed-integer program firstly optimizes the line-switching sequence and partial control resources. Then, two nonlinear programs recover AC feasibility, and optimize the remaining control resources by minimizing the $\mathcal{H}_2$-norm of associated linearized systems, respectively. The final transition scheme is selected by accurate evaluation including stability verification using time-domain simulations. Finally, numerical studies demonstrate the effectiveness and superiority of the proposed methodology to achieve bumpless topology transition.