T objectives of this review are to summarize and to evaluate the present state of knowledge concerning optimal control synthesis. The primary areas of interest are flight mechanics and flight control. The design of optimal stabilization and control systems, the determination of optimal flight paths, and the calculation of optimal orbital transfers have a common mathematical foundation in the calculus of variations; this review is limited to problems requiring a variational treatment. The selection of optimum vehicle shapes and the determination of optimum staging and propellant loadings for rockets are not included in the scope of this work although they are variational problems. A general discussion of optimization techniques draws upon the accumulated experience and research results of many workers in different branches of engineering and applied mathematics. In recent years, there has been a confluence of control research in aeronautical, electrical, mechanical, and chemical engineering. The research efforts in specialized technical fields have produced results of general interest, but the splintering of the literature according to old traditions has sometimes retarded the application of the results. A principal task of this review is to discuss the synthesis of closed-loop optimal controllers. This review will not make a sharp distinction between optimal guidance and optimal control because the mathematical problems are nearly identical even though the time scales of the system being controlled may differ by several orders of magnitude. A synthesis procedure for an optimal system is a set of design rules whose application results in a closed feedback loop comprised by the system being controlled and a computer to process the output measurements and to determine the optimal control law. This is illustrated in Fig. 1, which is a block diagram encompassing a very wide range of diverse problems extending from the optimal correction of interplanetary trajectories to optimal flight controllers for airplanes. These two apparently dissimilar technical problems are nevertheless nearly identical from a system synthesis viewpoint. The same basic design steps are required in each case. These include the selection of a performance criterion, the estimation of the statistical properties of noise and random inputs, the determination of the system state from noisy data, and the computation of the optimal control law. Each of these items will be taken up again later on.