  This article proposes a new approach based on finite-horizon parameterizing
manifolds (PMs) for the design of low-dimensional suboptimal controllers to
optimal control problems of nonlinear partial differential equations (PDEs) of
parabolic type. Given a finite horizon $[0,T]$ and a low-mode truncation of the
PDE, a PM provides an approximate parameterization of the uncontrolled high
modes by the controlled low ones so that the unexplained high-mode energy is
reduced, in an $L^2$-sense, when this parameterization is applied. Analytic
formulas of such PMs are derived by application of the method of pullback
approximation of the high-modes (Chekroun, Liu, and Wang, 2013,
http://arxiv.org/pdf/1310.3896v1.pdf). These formulas allow for an effective
derivation of reduced ODE systems, aimed to model the evolution of the low-mode
truncation of the controlled state variable, where the high-mode part is
approximated by the PM function applied to the low modes. A priori error
estimates between the resulting PM-based low-dimensional suboptimal controller
$u_R^\ast$ and the optimal controller $u^*$ are derived. These estimates
demonstrate that the closeness of $u_R^\ast$ to $u^*$ is mainly conditioned on
two factors: (i) the parameterization defect of a given PM, associated
respectively with $u_R^\ast$ and $u^*$; and (ii) the energy kept in the high
modes of the PDE solution either driven by $u_R^\ast$ or $u^*$ itself. The
practical performances of such PM-based suboptimal controllers are numerically
assessed for various optimal control problems associated with a Burgers-type
equation. The numerical results show that a PM-based reduced system allows for
the design of suboptimal controllers with good performances provided that the
associated parameterization defects and energy kept in the high modes are small
enough, in agreement with the rigorous results.
