This article focuses on sensitivity and control theory for linear complementarity systems (LCSs), a type of dynamical system that experiences hybrid continuous/discrete behavior and is therefore nonsmooth. In particular, a sensitivity theory is given that characterizes generalized derivative information of solutions of LCSs with respect to parametric perturbations. With this theory in hand, a computationally-relevant open-loop optimal control theory is provided using a direct method (i.e., the control is parametrically discretized and generalized gradients of the objective function are described). The approach here is based on lexicographic directional differentiation theory, a relatively new tool in nonsmooth analysis, being applied to nonlinear complementarity systems (NCSs). The optimal control theory is illustrated with an example. As a byproduct of the sensitivity theory, well-posedness results for a new class of hybrid dynamical system, called the lexicographic linear complementarity system (LexLCS), are also established.