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

Abstract

Abstract. This 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 [26]. 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.A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u * R and the optimal controller u * are derived under a second-order sufficient optimality condition. These estimates demonstrate that the closeness of u * R to u * is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with the suboptimal controller u * R and the optimal controller u * ; and (ii) the energy kept in the high modes of the PDE solution either driven by u * R 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.

Approximation of Stochastic Invariant Manifolds

“…Among the differences with the AIM approach, the PM approach seeks for manifolds which allows to provide-in a mean square sense-simple modeling error estimates for the evolution of u c , over any finite (sufficiently large) time interval; see [37,Proposition 5.1] and see also [34,Theorem 1 and Corollary 1]. This modeling error is controlled by the product of three terms: the energy of the unresolved modes (i.e., the unknown information), the nonlinear effects related to the size of the global random attractor, and the parameterization defect of the stochastic PM employed in the reduction.…”

“…Similarly, deterministic PMs can be defined and efficiently computed in the deterministic setting as briefly discussed in [37,Sect. 4.5] and further investigated in [34] for the design of low-dimensional suboptimal controllers of nonlinear parabolic PDEs.…”

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

“…to reduce a quantity like R(u * t,x , u N, * t,x N ) in (2.42) that involves the residual energies, Π ⊥ N y t,x (·; u * t,x ) L 2 (t,T;D(A)) and Π ⊥ N y t,x (·; u N, * t,x N ) L 2 (t,T;D(A)) . The theory of parameterizing manifolds (PM) [12,13,16] allows for such a reduction leading typically to approximate controls coming essentially with error estimates that introduce a multiplying factor 0 ≤ Q < 1 (related to the PM) in an RHS similar to that of (2.42); see [12,Theorem 1 & Corollary 2]. The combination of the GK framework of [9] with the PM reduction techniques of [12] constitutes thus an idea that is worth pursuing for solving efficiently optimal control problems of nonlinear DDEs.…”

“…In each case, the PMP approach leads to a boundary value problem (BVP) for the corresponding state and co-state variables, which is solved using the MATLAB built-in function bvp4c; see e.g. [12,Sect. 5] for more details.…”

Evolve Then Filter Regularization for Stochastic Reduced Order Modeling

“…The deterministic viscous Burgers equation and its stochastic version have been widely used in reduced order modeling, see [14,15,34,39,49]. In this paper, we will focus on the following stochastic Burgers equation (SBE) driven by linear multiplicative noise: du = νu xx − uu x dt + σu • dW t , u(0, t) = u(1, t) = 0, t ≥ 0, u(x, 0) = u 0 (x),…”