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