Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator - Peitz, Sebastian; Klus, Stefan (Lecture Notes in Control and Information Sciences)
In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.

Deep model predictive flow control with limited sensor data and online learning - Bieker, Katharina; Peitz, Sebastian; Brunton, Steven L.; Kutz, J. Nathan; Dellnitz, Michael (Theoretical and Computational Fluid Dynamics)
The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.

Data-Driven Model Predictive Control using Interpolated Koopman Generators - Peitz, Sebastian; Otto, Samuel E.; Rowley, Clarence W. (SIAM Journal on Applied Dynamical Systems)
In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.

A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors - Gerlach, Raphael; Ziessler, Adrian; Eckhardt, Bruno; Dellnitz, Michael (SIAM Journal on Applied Dynamical Systems)
In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes .

Variational integrators for dissipative systems - Limebeer, D. J. N.; Ober-Blöbaum, Sina; Farshi, F. H. (IEEE Transactions on Automatic Control)

Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty - Hernández Castellanos, Carlos Ignacio; Ober-Blöbaum, Sina; Peitz, Sebastian (International Journal of Robust and Nonlinear Control)
In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase, we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.

Numerical computation of lightly multi-objective robust optimal solutions by means of generalized cell mapping - Hernández Castellanos, C. I. O.; Schütze, G.; Sun, J.-Q.; Ober-Blöbaum, Sina; Morales-Luna, G. (Mathematics)

A multirate variational approach to simulation and optimal control for flexible spacecraft - Lishkova, Y.; Ober-Blöbaum, Sina; Cannon, M.; Leyendecker, S. (Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe)

A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems - Faulwasser, T.; Flaßkamp, K.; Ober-Blöbaum, Sina; Worthmann, K. (24th International Symposium on Mathematical Theory of Networks and Systems)

Modelling of the convection-diffusion equation through fractional restricted calculus of variations - Cresson, J.; Jiménez, F.; Ober-Blöbaum, Sina (24th International Symposium on Mathematical Theory of Networks and Systems)

On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry - Jean, Frédéric; Maslovskaya, Sofya; Zelenko, Igor (Geometriae Dedicata)
Keywords: Geometry and Topology

The turnpike property in maximization of microbial metabolite production - Maslovskaya, Sofya; Caillau, Jean-Baptiste; Djema, Walid; Giraldi, Laetitia; Jean-Luc, Jean-Luc; Pomet, Jean-Baptiste

Imaginary quadratic number fields with class groups of small exponent - Elsenhans, Andreas-Stephan; Klüners, Jürgen; Nicolae, Florin (Acta Arithmetica)
Let D<0 be a fundamental discriminant and denote by E(D) the exponent of the ideal class group Cl(D) of K=ℚ(√D). Under the assumption that no Siegel zeros exist we compute all such D with E(D) dividing 8. We compute all D with |D| ≤ 3.1⋅10²⁰ such that E(D) ≤ 8.
Keywords: Algebra and Number Theory

The conductor density of local function fields with abelian Galois group - Klüners, Jürgen; Müller, Raphael (Journal of Number Theory)
We give an exact formula for the number of G-extensions of local function fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
Keywords: Algebra and Number Theory

Symmetry in optimal control: A multiobjective model predictive control approach - Flaßkamp, K.; Ober-Blöbaum, Sina; Peitz, S. (Advances in Dynamics, Optimization and Computation)

The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems - Gerlach, Raphael; Ziessler, Adrian (Advances in Dynamics, Optimization and Computation)
In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.

On the equivariance properties of self-adjoint matrices - Dellnitz, Michael; Gebken, Bennet; Gerlach, Raphael; Klus, Stefan (Dynamical Systems)
We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.

Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates - Hipp, David; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates - Hipp, David; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation - Akrivis, Georgios; Feischl, Michael; Kovács, Balázs; Lubich, Christian (Mathematics of Computation)
Keywords: Applied Mathematics; Computational Mathematics; Algebra and Number Theory