  Optimal controller synthesis is a bilinear problem and hence difficult to
solve in a computationally efficient manner. We are able to resolve this
bilinearity for systems with delay by first convexifying the problem in
infinite-dimensions - formulating the $H_\infty$ optimal state-feedback
controller synthesis problem for distributed-parameter systems as a Linear
Operator Inequality - a form of convex optimization with operator variables.
Next, we use positive matrices to parameterize positive `complete quadratic'
operators - allowing the controller synthesis problem to be solved using
Semidefinite Programming (SDP). We then use the solution to this SDP to
calculate the feedback gains and provide effective methods for real-time
implementation. Finally, we use several test cases to verify that the resulting
controllers are \textit{optimal} to several decimal places as measured by the
minimal achievable closed-loop $H_\infty$ norm, and as compared against
controllers designed using high-order Pad\'e approximations.
