  This work presents a convex-optimization-based framework for analysis and
control of nonlinear partial differential equations. The approach uses a
particular weak embedding of the nonlinear PDE, resulting in a linear equation
in the space of Borel measures. This equation is then used as a constraint of
an infinite-dimensional linear programming problem (LP). This LP is then
approximated by a hierarchy of convex, finite-dimensional, semidefinite
programming problems (SDPs). In the case of analysis of uncontrolled PDEs, the
solutions to these SDPs provide bounds on a specified, possibly nonlinear,
functional of the solutions to the PDE; in the case of PDE control, the
solutions to these SDPs provide bounds on the optimal value of a given optimal
control problem as well as suboptimal feedback controllers. The entire approach
is based purely on convex optimization and does not rely on spatio-temporal
gridding, even though the PDE addressed can be fully nonlinear. The approach is
applicable to a very broad class nonlinear PDEs with polynomial data.
Computational complexity is analyzed and several complexity reduction
procedures are described. Numerical examples demonstrate the approach.
