This paper considers linear time-varying hybrid systems and introduces a notion of the finite-horizon generalized H2 norm with transients. It is defined as the worst-case peak value of the output in response to uncertain initial states and external disturbances. Such a measure represents the induced operator norm from L2 to L∞ because the peak value of the vector signal is considered as its generalized L∞ norm. This approach allows to characterize the generalized H2 norm in terms of both the difference Lyapunov equation and diference linear matrix inequalities (DLMIs). By using the derived characterization the optimal control and Pareto optimal controls are synthesized minimizing the finite-horizon generalized H2 norm with transients. Finally, an example is given to illustrate the proposed technique.