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In this paper, we present a dual formulation for Lyapunov's stability testand Bounded-real Lemma for Partial Integral Equations (PIEs). Then, we use thisformulation to solve the $H_{\infty}$-optimal controller synthesis problem forODE-PDEs that can be converted to a PIE. First, we provide a general dualcriterion for the Lyapunov stability and the Bounded-real Lemma for PIEs usingquadratic Lyapunov functionals. Then, we use a class of operators calledPartial Integral (PI) operators to parametrize the said Lyapunov functional andexpress the dual criterion as an operator-valued convex optimization problemwith sign-definite constraints on PI operators -- called Linear PI Inequalities(LPIs). Consequently, the optimal controller synthesis problem for PIEs, andthus, for PDEs, is formulated as LPIs. Finally, the LPI optimization problemsare solved using a computational toolbox called PIETOOLS and tested on variousexamples to demonstrate the utility of the proposed method.

$H_{\infty}$-optimal control of coupled ODE-PDE systems using PIE framework and LPIs