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

摘要: This article proposes a new approach for the design of low-dimensional suboptimal controllers to optimal control problems nonlinear partial differential equations (PDEs) parabolic type. The fits into long tradition seeking slaving relationships between small scales and large ones (to be controlled) but differ by introduction type manifolds do so, namely finite-horizon parameterizing (PMs). Given finite horizon [0,T] low-mode truncation PDE, PM provides an approximate parameterization high modes controlled low so that unexplained high-mode energy is reduced--in mean-square sense over [0,T]--when this applied. Analytic formulas such PMs are derived application method pullback approximation high-modes introduced in Chekroun et al. (2014). These allow effective derivation reduced systems ordinary (ODEs), aimed model evolution state variable, where part approximated function applied modes. then obtained (indirect) techniques from finite-dimensional theory, PM-based ODEs. A priori error estimates resulting controller $u_{R}^{\ast}$ u? under second-order sufficient optimality condition. demonstrate closeness mainly conditioned on two factors: (i) defect given PM, associated respectively with u?; (ii) kept PDE solution either driven or itself. The practical performances numerically assessed Burgers-type equation; locally as well globally distributed cases being both considered. numerical results show system allows good provided defects enough, agreement rigorous results.