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Electrochemical reaction processes attract increasing attention as a promising chemical process alternative to achieve green and sustainable chemical manufacturing due to its property that can produce chemical products without directly burning fossil fuels. Among various electrochemical reaction processes, CO\textsubscript{2} reduction provides a possibility to capture the CO\textsubscript{2} gas in the atmosphere and effectively relieve the global warming crisis. However, due to the complex, stochastic, and nonlinear nature of the electrochemical reactions, modeling an electrochemical process using first-principles is very challenging. To investigate the physical-chemical phenomena of the electrochemistry of reduction of CO\textsubscript{2}, the Laboratory of Electrochemical System Engineering at UCLA has developed a gastight rotating cylinder electrode (RCE) cell to decouple mass transport phenomena from the intrinsic kinetics of the electrochemical reactions. Specifically, this RCE cell allows manipulation of two variables, the applied potential and rotation speed of the cylindrical electrode, which determines the mass transport profile and reaction kinetics of the electrochemical reactions, respectively. This design further enables the development of an advanced process control system for the reactor to control the desired output states by adjusting the two key input variables.However, the absence of the first-principle model for the electrochemical reactor is a significant challenge to develop the process control system. To this end, considering the complexity of the process, neural network (NN) models are developed and used in this thesis work to capture the input-output relationship of the reactor and provide a data-based alternative to the unavailable, first-principle models. On the other hand, in a data-driven approach, the performance of the NN model is determined by the quality of the training data. Therefore, the inevitable noise, caused by experimental uncertainty, can corrupt the collected data and negatively impact the NN modeling task. Motivated by this concern, various methods, such as the Monte Carlo dropout and co-teaching algorithm, are adopted in this thesis to improve the robustness of the NN models against noisy data.Eventually, this dissertation demonstrates an advanced model predictive (MPC) scheme based on experimental data-driven NN models that capture the input-output relationship of the electrochemical reactor. Furthermore, the Koopman operator method is adopted to perform on-line linearization to the NN model to improve the computational efficiency of the MPC and enable its real-time application to the experimental electrochemical reactor. Finally, simulations, open-, and closed-loop experiments are conducted to demonstrate the overall implementation and successful performance of the proposed NN models and MPC schemes.

During the 1930s, researchers realized that an abstract dynamical system induces a group of linear operators acting on the space of square-integrable functions. For measure-preserving systems, the induced operator is unitary and self-adjoint. As such, its spectrum is restricted to the unit circle and has been shown to encode many important statistical and geometric properties of the dynamical system. Since then, the induced Koopman group of operators' spectral properties have drawn an immense amount of research interest over the last decade. Due to the rise of computing capabilities and data availability, there has been an explosive amount of research into developing data-driven algorithms that can compute the spectrum numerically from data. In the first part of this dissertation, we demonstrate how Koopman operator methods can offer a model-free, data-driven approach to analyze and forecast highway traffic dynamics. By obtaining a decomposition of data sets collected by the Federal Highway Administration and the California Department of Transportation, we can reconstruct observed data, distinguish any growing or decaying patterns, and obtain a hierarchy of previously identified and never before identified spatiotemporal patterns. Furthermore, it is demonstrated how this methodology can be utilized to forecast highway network conditions. The developed forecasting scheme readily generalizes to the much-needed scenario of multi-lane highway networks without any loss to its performance or efficiency. Also, we do not rely on large historical training data nor parameter tuning or selection. Thereby providing a completely efficient and accurate method of analyzing and forecasting traffic patterns at the levels required by modern intelligent transportation systems.In the second part of this dissertation, we consider the equivalent induced linear operators acting on the space of sections of the tangent, cotangent, and tensor bundles of the state space. We begin by first demonstrating how these operators are indeed natural generalizations of Koopman operators acting on functions. The fundamental insight lies in understanding the connection between the differential geometric concept of pulling back objects (functions, vector fields, covector fields, tensor fields) under a diffeomorphism and how their pull-back relates to the Lie derivative of that object. We then draw connections between the various operators' spectrum and characterize the algebraic and differential topological properties of their spectrum. We describe these operators' discrete spectrum for linear dynamical systems and derive spectral type expansions for linear vector fields. The expansions derived resemble the familiar spectral expansion of functions under the Koopman operator. We define the notion of an "eigendistribution" and provide conditions for when an eigendistribution is integrable. We then demonstrate how to recover the foliations arising from their integral manifolds via the level sets of Koopman eigenfunctions. Many of the results presented in the second part of this dissertation stem from well-known differential geometric concepts. Prior work on such generalized operators on vector fields exists but has remained mostly unnoticed by the growing Koopman operator community. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized. Specifically, we show that the flows of two vector fields commute, subject to an appropriate rescaling of the flow time, if and only if one vector field is an eigensection of the other vector field. The eigenvalue prescribes the required time scaling, and we recover the original statement that the flows of commuting vector fields commute as a particular case of our result. We also apply our results to the study of a hyperbolic toral automorphism known as Arnold's Cat map. We demonstrate that the Lyapunov exponents are contained within the spectrum of the induced operator on vector fields, and we recover the stable and unstable foliations via the level sets of the joint eigenfunctions of the stable and unstable eigendistributions.

This article presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multistep prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control (MPC) framework of (M. Korda and I. Mezic, 2018) to control nonlinear dynamical systems using linear MPC tools. The method is entirely data-driven and based predominantly on convex optimization. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples demonstrate the approach, both for prediction and feedback control. * * Code for the numerical examples is available from https://homepages.laas.fr/mkorda/Eigfuns.zip .

This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis and the global, operator theoretic, Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link between the geometric theory and the transfer operator theory., Comment: 30 pages, 4 figures

In the paper, we study the so-called p-safety of a random dynamical system. We generalize the existing results for safety barrier certificates for deterministic dynamical systems and Markov processes. Moreover, we consider the case of random obstacles, modelled as random sets. This leads to the necessity of using integrals with respect to lower and upper distributions. We prove that if there exists at least one barrier certificate then the random dynamical system is safe. The barrier certificates are also defined using such nonlinear distributions. Furthermore, when the family of stochastic Koopman operators has the semigroup property, the barrier certificates are solutions for some type of Dirichlet problems.

Nonlinear biochemical systems such as the anaerobic digestion process experience the problem of the multi-stability phenomena, and thus, the dynamic spectrum of the system has several undesired equilibrium states. As a result, the selection of initial conditions and operating parameters to avoid such states is of importance. In this work, we present a data-driven approach, which relies on the generation of several system trajectories of the anaerobic digestion system and the construction of a data-driven Koopman operator to give a concise criterion for the classification of arbitrary initial conditions in the state space. Unlike other approximation methods, the criterion does not rely on difficult geometrical analysis of the identified boundaries to produce the classification.

The dominating methodology used in the study of dynamical systems is the geometric picture introduced by Poincare. The focus is on the structure of the state space and the asymptotic behavior of trajectories. Special solutions such as fixed points and limit cycles, along with their stable and unstable manifolds, are of interest due to their ability to organize the trajectories in the surrounding state space.Another viewpoint that is becoming more prevalent is the operator-theoretic / functional-analytic one which describes the system in terms of the evolution of functions or measures defined on the state space. Part I of this doctoral dissertation focuses on the Koopman, or composition, operator that determines how a function on the state space evolves as the state trajectories evolve. Most current studies involving the Koopman operator have dealt with its spectral properties that are induced by dynamical systems that are, in some sense, stationary (in the probabilistic sense). The dynamical systems studied are either measure-preserving or initial conditions for trajectories are restricted to an attractor for the system. In these situations, only the point spectrum on the unit circle is considered; this part of the spectrum is called the unimodular spectrum. This work investigates relaxations of these situations in two different directions. The first is an extension of the spectral analysis of the Koopman operator to dynamical systems possessing either dissipation or expansion in regions of their state space. The second is to consider switched, stochastically-driven dynamical systems and the associated collection of semigroups of Koopman operators.In the first direction, we develop the Generalized Laplace Analysis (GLA) for both spectral operators of scalar type (in the sense of Dunford) and non spectral operators. The GLA is a method of constructing eigenfunctions of the Koopman operator corresponding to non-unimodular eigenvalues. It represents an extension of the ergodic theorems proven for ergodic, measure-preserving, on-attractor dynamics to the case where we have off-attractor dynamics. We also give a general procedure for constructing an appropriate Banach space of functions on which the Koopman operator is spectral. We explicitly construct these spaces for attracting fixed points and limit cycles. The spaces that we introduce and construct are generalizations of the familiar Hilbert Hardy spaces in the complex unit disc.In the second direction, we develop the theory of switched semigroups of Koopman operators. Each semigroup is assumed to be spectral of scalar-type with unimodular point spectrum, but possibly non-unimodular continuous spectrum. The functions evolve by applying one semigroup for a period of time, then switching to another semigroup. We develop an approximation of the vector-valued function evolution by a linear approximation in the vector space that the functions map into. A basis for this linear approximation is constructed from the vector-valued modes that are coefficients of the projections of the vector-valued observable onto scalar-valued eigenfunctions of the Koopman operator. The unmodeled modes show up as noisy dynamics in the output space. We apply this methodology to traffic matrices of an Internet Service Provider's (ISP's) network backbone. Traffic matrices measure the traffic volume moving between an ingress and egress router for the network's backbone. It is shown that on each contiguous interval of time in which a single semigroup acts the modal dynamics are deterministic and periodic with Gaussian or nearly-Gaussian noise superimposed.Part II of the dissertation represents a divergence from the first part in that it does not deal with the Koopman operator explicitly. In the second part, we consider the problem of using exponentially mixing dynamical systems to generate trajectories for an agent to follow in its search for a physical target in a large domain. The domain is a compact subset of the n-dimensional Euclidean space Rn. It is assumed that the size of the target is unknown and can take any value in some continuous range. Furthermore, it is assumed that the target can be located anywhere in the domain with equal probability.We cast this problem as one in the field of quantitative recurrence theory, a relatively new sub-branch of ergodic theory. We give constructive proofs for upper bounds of hitting times of small metric balls in Rn for mixing transformations of various speeds. The upper bounds and limit laws we derive say, approximately, that the hitting time is bounded above by some constant multiple of the inverse of the measure of the metric ball. From these results, we derive upper bounds for the expected hitting time, with respect to the range of target sizes [delta, V), to be of order O(-ln delta). First order, continuous time dynamics are constructed from discrete time mixing transformations and upper bounds for these hitting times are shown to be proportional to the discrete time case.

Hierarchical support vector regression (HSVR) models a function from data as a linear combination of SVR models at a range of scales, starting at a coarse scale and moving to finer scales as the hierarchy continues. In the original formulation of HSVR, there were no rules for choosing the depth of the model. In this paper, we observe in a number of models a phase transition in the training error -- the error remains relatively constant as layers are added, until a critical scale is passed, at which point the training error drops close to zero and remains nearly constant for added layers. We introduce a method to predict this critical scale a priori with the prediction based on the support of either a Fourier transform of the data or the Dynamic Mode Decomposition (DMD) spectrum. This allows us to determine the required number of layers prior to training any models., 18 pages, 9 figures

Stochastic parameterizations are increasingly being used in climate modeling to represent subgrid‐scale processes. While different parameterizations are being developed considering different aspects of the physical phenomena, less attention is given to technical and numerical aspects. In particular, empirical orthogonal functions (EOFs) are employed when a spatial structure is required. Here, we provide evidence they might not be the most suitable choice. By applying an energy‐consistent parameterization to the two‐layer quasi‐geostrophic (QG) model, we investigate the model sensitivity to a priori assumptions made on the parameterization. In particular, we consider here two methods to prescribe the spatial covariance of the noise: first, by using climatological variability patterns provided by EOFs, and second, by using time‐varying dynamics‐adapted Koopman modes, approximated by dynamic mode decomposition (DMD). The performance of the two methods are analyzed through numerical simulations of the stochastic system on a coarse spatial resolution and the outcomes compared to a high‐resolution simulation of the original deterministic system. The comparison reveals that the DMD‐based noise covariance scheme outperforms the EOF‐based one. The use of EOFs leads to a significant increase of the ensemble spread and to a meridional misplacement of the bimodal eddy kinetic energy (EKE) distribution. Conversely, using DMDs, the ensemble spread is confined, the meridional propagation of the zonal jet stream is accurately captured, and the total variance of the system is improved. Our results highlight the importance of the systematic design of stochastic parameterizations with dynamically adapted spatial correlations, rather than relying on statistical spatial patterns.

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is founded on the density function based almost everywhere stability certificate that is dual to the Lyapunov function for dynamic systems. Unlike Lyapunov based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently by invoking the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory of the dynamical system can be understood using linear Perron-Frobenius and Koopman operators. This connection allows us to use data-driven methods developed to approximate these operators combined with the SOS techniques for the convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples., 6 pages, 5 figures

We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov’s formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators.

This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theoretical result on uniform boundedness of the invariant sets is presented. The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our theoretical results can be used to understand stability regions in such complex systems., 23 pages, 4 figures

We exploit the key idea that nonlinear system identification is equivalent to linear identification of the socalled Koopman operator. Instead of considering nonlinear system identification in the state space, we obtain a novel linear identification technique by recasting the problem in the infinite-dimensional space of observables. This technique can be described in two main steps. In the first step, similar to the socalled Extended Dynamic Mode Decomposition algorithm, the data are lifted to the infinite-dimensional space and used for linear identification of the Koopman operator. In the second step, the obtained Koopman operator is "projected back" to the finite-dimensional state space, and identified to the nonlinear vector field through a linear least squares problem. The proposed technique is efficient to recover (polynomial) vector fields of different classes of systems, including unstable, chaotic, and open systems. In addition, it is robust to noise, well-suited to model low sampling rate datasets, and able to infer network topology and dynamics., 6 pages

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity., Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office, This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jsv.2016.05.016

We completely describe the weak closure of the powers of the Koopman operator associated with Chacon’s classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of$\alpha $-weak mixing for Chacon’s automorphism.

This article is concerned with conjugacy problems arising in homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms., Comment: 23 pages