This paper considers the design of minimax and optimal guaranteed cost controllers for uncertain linear systems. The uncertainty is permitted to have a structured description and is bounded by an integral quadratic constraint. For known initial conditions on the system state, it is shown that minimax control can be realised by static full state feedback controllers. A new proof of minimax optimality is given using standard methods for linear systems. The design of such controllers requires the optimal solution of a single Riccati equation dependent upon a set of parameters whose dimension is equal to the number of uncertainty blocks. It is shown that this multivariable optimisation is convex and thus numerically tractable. No explicit criteria to determine the existence of a solution are known; however, if solutions exist, results pertaining to their location are given to assist numerical optimisation. The extension of the approach to arbitrary unknown initial conditions is considered but is shown, in general, not to be possible. A tractable approach, considering the initial condition to be a random variable whose occurrence satisfies a certain probability density function, is presented. The resulting optimisation problem is shown to be convex and thus numerically tractable.