Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs

We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.Comment: 29 pages, 4 figure

Practical asymptotic stability of data-driven model predictive control using extended DMD

The extended Dynamic Mode Decomposition (eDMD) is a very popular method to obtain data-driven surrogate models for nonlinear (control) systems governed by ordinary and stochastic differential equations. Its theoretical foundation is the Koopman framework, in which one propagates observable functions of the state to obtain a linear representation in an infinite-dimensional space. In this work, we prove practical asymptotic stability of a (controlled) equilibrium for eDMD-based model predictive control, in which the optimization step is conducted using the data-based surrogate model. To this end, we derive error bounds that converge to zero if the state approaches the desired equilibrium. Further, we show that, if the underlying system is cost controllable, then this stabilizablility property is preserved. We conduct numerical simulations, which illustrate the proven practical asymptotic stability.Comment: 25 pages, 5 figure

Error bounds for kernel-based approximations of the Koopman operator

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.Comment: 28 page