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In this paper, results for quadratic optimal control of hybrid linear autonomous systems with state jumps are reported. In particular, we focus on problems in which a pre-specified sequence of active subsystems is given. In a recent paper, we have proposed an approach for optimal control problems of hybrid autonomous systems with state jumps. However, in applying that approach, some computational issues must be addressed for the derivation of some necessary parameters. We, in this paper, by taking advantage of the special structure of linear subsystems and quadratic costs, develop a more efficient method which computes these necessary parameters and hence leads to reduced computational effort. As an application, we apply our quadratic optimal control results to address the important reachability problems. Examples illustrate the results.