  In this paper, we present solvable, convex formulations of $H_2$-optimal
state estimation and state-feedback control problems for a general class of
linear Partial Differential Equations (PDEs) with one spatial dimension. These
convex formulations are derived by using an analysis and control framework
called the `Partial Integral Equation' (PIE) framework, which utilizes the PIE
representation of infinite-dimensional systems. Since PIEs are parameterized by
Partial Integral (PI) operators that form an algebra, $H_2$-optimal estimation
and control problems for PIEs can be formulated as Linear PI Inequalities
(LPIs). Furthermore, if a PDE admits a PIE representation, then the stability
and $H_2$ performance of the PIE system implies that of the PDE system.
Consequently, the optimal estimator and controller obtained for a PIE using
LPIs provide the same stability and performance when applied to the
corresponding PDE. These LPI optimization problems can be solved
computationally using semi-definite programming solvers because such problems
can be formulated using Linear Matrix Inequalities by using positive matrices
to parameterize a cone of positive PI operators. We illustrate the application
of these methods by constructing observers and controllers for some standard
PDE examples.
