In this paper, we present a convex formulation of H8 -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 H8 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 H8 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 H8 -performance problems using the Linear PI Inequality (LPI) framework. Next, we use our duality results to formulate the stabilization and H8 - optimal state-feedback control problems as LPIs. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.