  In this paper, we present a convex formulation of $H_{\infty}$-optimal
control problem for coupled linear ODE-PDE systems with one spatial dimension.
First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation
(PIE) system and show that stability and $H_{\infty}$ performance of the PIE
system implies that of the ODE-PDE system. We then construct a dual PIE system
and show that asymptotic stability and $H_{\infty}$ performance of the dual
system is equivalent to that of the primal PIE system. Next, we pose a convex
dual formulation of the stability and $H_{\infty}$-performance problems using
the Linear PI Inequality (LPI) framework. LPIs are a generalization of LMIs to
Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB
toolbox. Next, we use our duality results to formulate the stabilization and
$H_{\infty}$-optimal state-feedback control problems as LPIs. Finally, we
illustrate the accuracy and scalability of the algorithms by constructing
controllers for several numerical examples.
