Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and nonlinear Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes- Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.Comment: 11 page

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

Fast and memory-efficient optimization for large-scale data-driven predictive control

Recently, data-enabled predictive control (DeePC) schemes based on Willems' fundamental lemma have attracted considerable attention. At the core are computations using Hankel-like matrices and their connection to the concept of persistency of excitation. We propose an iterative solver for the underlying data-driven optimal control problems resulting from linear discrete-time systems. To this end, we apply factorizations based on the discrete Fourier transform of the Hankel-like matrices, which enable fast and memory-efficient computations. To take advantage of this factorization in an optimal control solver and to reduce the effect of inherent bad conditioning of the Hankel-like matrices, we propose an augmented Lagrangian lBFGS-method. We illustrate the performance of our method by means of a numerical study