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Optimal controller synthesis is a bilinear problem and hence difficult tosolve in a computationally efficient manner. We are able to resolve thisbilinearity for systems with delay by first convexifying the problem ininfinite-dimensions - formulating the $H_\infty$ optimal state-feedbackcontroller synthesis problem for distributed-parameter systems as a LinearOperator 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 usingSemidefinite Programming (SDP). We then use the solution to this SDP tocalculate the feedback gains and provide effective methods for real-timeimplementation. Finally, we use several test cases to verify that the resultingcontrollers are \textit{optimal} to several decimal places as measured by theminimal achievable closed-loop $H_\infty$ norm, and as compared againstcontrollers designed using high-order Pad\'e approximations.

A Convex Solution of the $H_\infty$-Optimal Controller Synthesis Problem for Multi-Delay Systems