This paper considers the design of minimax controllers for a class of linear systems. The system is perturbed by a known initial condition on the state and by a disturbance process constrainted by an L-2 norm bound. Structured undertainty is represented by perturbation feedback through L-2-induced norm bounded operators. The performance is measured by an integral quadratic cost function. The resultant controllers minimise the worst-case cost for all admissible disturbances and for all admissible systems. The design involves the optimal solution of a parametric algebraic Riccati equation. A new proof of minimax optimality is given using standard methods for linear systems. The multivariable optimisation problem is shown to be convex and the set of feasible parameter values is shown to be compact. Thus, the design procedure is computationally tractable.