In this paper, performance analysis for a class of hybrid systems and optimization-based synthesis of a multi-objective reset controller for linear plants are presented. Lyapunov-based conditions to estimate L2 gain bounds generalizing and relaxing previous results in the literature are provided in an analysis context. In particular, the analysis results allow for growth of the Lyapunov function at jumps, leading to numerically tractable conditions. On the other hand, the synthesis result allows designing via convex tools with a line search a multi-objective reset controller optimizing both the exponential converging rate and the L2 gain. A novel peculiarity of our scheme is that the underlying linear flow dynamics is not necessarily stabilizing. This last property is the consequence of taking into account in the design where the flowing trajectories lay and leads to improved performance, especially in terms of decay rate. Some simulations illustrate the usefulness of our techniques.