Honghu Liu

Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations

General Introduction.- Preliminaries.- Invariant Manifolds.- Pullback Characterization of Approximating, and Parameterizing Manifolds.- Non-Markovian Stochastic Reduced Equations.- On-Markovian Stochastic Reduced Equations on the Fly.- Proof of Lemma 5.1.-References.- Index.

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

AbstractThis article proposes a new approach for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. The approach fits into the long tradition of seeking for slaving relationships between the small scales and the large ones (to be controlled) but differ by the introduction of a new type of manifolds to do so, namely the finite-horizon parameterizing manifolds (PMs). Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the high modes by the controlled low ones so that the unexplained high-mode energy is reduced—in a mean-square sense over [0,T]—when this parameterization is applied.Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes introduced in Chekroun et al. (2014). These formulas allow for an effective derivation of reduced systems of ordinary differential equations (ODEs), 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. The design of low-dimensional suboptimal controllers is then obtained by (indirect) techniques from finite-dimensional optimal control theory, applied to the PM-based reduced ODEs.n A priori error estimates between the resulting PM-based low-dimensional suboptimal controller

Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility

u_{R}^{ast}

Post-processing finite-horizon parameterizing manifolds for optimal control of nonlinear parabolic PDEs

and the optimal controller u∗ are derived under a second-order sufficient optimality condition. These estimates demonstrate that the closeness of

Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

u_{R}^{ast}

Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II

to u∗ is mainly conditioned on two factors: (i)xa0the parameterization defect of a given PM, associated respectively with the suboptimal controller

Approximation of stochastic invariant manifolds : stochastic manifolds for nonlinear SPDEs I

u_{R}^{ast}

Approximation of Stochastic Invariant Manifolds

and the optimal controller u∗; and (ii)xa0the energy kept in the high modes of the PDE solution either driven by

LOW-DIMENSIONAL GALERKIN APPROXIMATIONS OF NONLINEAR DELAY DIFFERENTIAL EQUATIONS

u_{R}^{ast}

The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories

or u∗ itself.The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. 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.