Frobenius-Perron Operators

Uncertainty propagation is an important step in the derivation of optimal control strategies for dynamic systems in the presence of state and parameter uncertainty. Many stochastic control formulations seek to optimize an expected value of a score or cost function, or otherwise enforce a probabilistic constraint through the use of an expected value. The time-varying state density needed for this expected value computation may be quantified through Monte Carlo methods or explicit approaches such as the Frobenius-Perron operator. This paper explores an alternative approach to the computation of the expected value by leveraging the adjoint relationship between the Frobenius-Perron operator and the Koopman operator. This relationship states that the expected value of a forward-propagated density with a score function is equivalent to the expected value of the initial density with the score function mapped back to the initial state space. By computing the expected value through use of the Koopman operator, several computational issues arising when using the Frobenius-Perron operator can be mitigated, while maintaining the benefits of an explicit method over Monte Carlo. A general stochastic optimal control problem is formulated wherein the expected value is computed between the initial state density and a score function using the Koopman operator. The computational benefits of computing the expected value through use of the Koopman operator are discussed, particularly with respect to numerical issues encountered using explicit forward propagation of the state probability density. A series of numerical examples highlight these computational advantages and illustrate the benefit of using this adjoint approach in solution to a range of optimal control problems in the presence of uncertainty.

We provide a data-driven framework for optimal control of a continuous-time control-affine stochastic dynamical system. The proposed framework relies on the linear operator theory involving Perron-Frobenius (P-F) and Koopman operators. Our first result involving the P-F operator provides a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the linear operators. We provide rigorous error bounds and convergence rate for the data-driven approximation of the linear operators and comment on the convergence rate of the optimal control w.r.t. data size. Our second results demonstrate using the Koopman operator, dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving the P-F operator and the Koopman-based formulation using the HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of the P-F and Koopman operators. Finally, we present examples to demonstrate the efficacy of the developed framework and verify the convergence rates for the operator and optimal control numerically.

In recent years data-driven analysis of dynamical systems has attracted a lot of attention and transfer operator techniques, namely, Perron-Frobenius and Koopman operators are being used almost ubiquitously. Since data is always obtained in discrete-time, in this paper, we propose a purely data-driven approach for the design of a stabilizing feedback control law for a general class of discrete-time control-affine non-linear systems. In particular, we use the Koopman operator to lift a control-affine system to a higher-dimensional space, where the control system's evolution is bilinear. We analyze the controllability of the lifted bilinear system and relate it to the controllability of the underlying non-linear system. We then leverage the concept of Control Lyapunov Function (CLF) to design a state feedback law that stabilizes the origin. Furthermore, we demonstrate the efficacy of the proposed method to stabilize the origin of the Van der Pol oscillator and the chaotic Henon map from the time-series data.

This paper introduces a framework for solving time-autonomous nonlinear infinite horizon optimal control problems, under the assumption that all minimizers satisfy Pontryagin’s necessary optimality conditions. In detail, we use methods from the field of symplectic geometry to analyze the eigenvalues of a Koopman operator that lifts Pontryagin’s differential equation into a suitably defined infinite dimensional symplectic space. This has the advantage that methods from the field of spectral analysis can be used to characterize globally optimal control laws. A numerical method for constructing optimal feedback laws for nonlinear systems proceeds by computing the eigenvalues and eigenvectors of a matrix that is obtained by projecting the Pontryagin–Koopman operator onto a finite dimensional space. We illustrate the effectiveness of this approach by computing accurate approximations of the optimal nonlinear feedback law for a Van der Pol control system, which cannot be stabilized by a linear control law.

Multivariable time series forecasting methods can integrate information from exogenous variables, leading to significant prediction accuracy gains. The transformer architecture has been widely applied in various time series forecasting models due to its ability to capture long-range sequential dependencies. However, a naïve application of transformers often struggles to effectively model complex relationships among variables over time. To mitigate against this, we propose a novel architecture, termed Spectral Operator Neural Network (Sonnet). Sonnet applies learnable wavelet transformations to the input and incorporates spectral analysis using the Koopman operator. Its predictive skill relies on the Multivariable Coherence Attention (MVCA), an operation that leverages spectral coherence to model variable dependencies. Our empirical analysis shows that Sonnet yields the best performance on 34 out of 47 forecasting tasks with an average mean absolute error (MAE) reduction of 2.2% against the most competitive baseline. We further show that MVCA can remedy the deficiencies of naïve attention in various deep learning models, reducing MAE by 10.7% on average in the most challenging forecasting tasks.

Computer programs for ESR spectral analysis

We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.

Vitamin K-dependent carboxylase: Evidence for cofractionation of carboxylase and epoxidase activities, and for carboxylation of a high-molecular-weight microsomal protein

The problem of an aperture-coupled patch antenna is studied using the spectral domain technique. The general case of a slot inclined to both the feedline and the patch is considered. The equivalent circuit for the structure is obtained and the effects of various parameters, such as slot dimensions and inclination on the resonant frequency and on the input impedance, are studied. Comparison between theory and measurement shows good agreement. The analysis could be useful in designing antennas for dual frequency operation or for circular polarisation where inclined slots could be used.

This paper considers the problem of representing a sufficiently smooth nonlinear dynamical as a structured potential-driven system. The proposed approach is based on a decomposition of a differential one-form that encodes the divergence of the given vector fields into its exact and anti-exact components, and into its co-exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems, and generalized Hamiltonian systems are given to illustrate the proposed approach.

Spectral decomposition of dynamical systems is a popular methodology to investigate the fundamental qualitative and quantitative properties of these systems and their solutions. In this chapter, we consider a class of nonlinear cooperative protocols, which consist of multiple agents that are coupled together via an undirected state-dependent graph. We develop a representation of the system solution by decomposing the nonlinear system utilizing ideas from the Koopman operator theory and its spectral analysis. We use recent results on the extensions of the well-known Hartman theorem for hyperbolic systems to establish a connection between the original nonlinear dynamics and the linearized dynamics in terms of Koopman spectral properties. The expected value of the output energy of the nonlinear protocol, which is related to the notions of coherence and robustness in dynamical networks, is evaluated and characterized in terms of Koopman eigenvalues, eigenfunctions, and modes. Spectral representation of the performance measure enables us to develop algorithmic methods to assess the performance of this class of nonlinear dynamical networks as a function of their graph topology. Finally, we propose a scalable computational method for approximation of the components of the Koopman mode decomposition, which is necessary to evaluate the systemic performance measure of the nonlinear dynamic network.

Deformation analysis of laser spectra based on S-R decomposition theorem

By studying the strong uniquess properties of best complex Chebyshev approximations to perturbations of a special class of functions, we show that strong uniqueness is not a generic property when we restrict ourselves to nonoscillatory'' functions. 10 refs.

This study tackles the issue of electromechanical modes identification through a measurement‐based methodology employing a novel signal decomposition theorem based upon the Hilbert transform. The methodology aims to answer in a simpler and more pragmatic manner to the main weaknesses of the Hilbert–Huang transform with respect to the major refinements in the relevant literature. These weak points are discussed with sufficient detailed degree in the study. The main contribution of this study consists in combining a recent signal decomposition theorem for separating an assigned signal into elemental ones, each of them characterised by a single frequency component and a robust preliminary non‐linear spectral analyser, named L p periodogram. This procedure's results are very appropriate for analysing some critical cases of electromechanical oscillations, because of the L p periodogram robustness against heavy‐tailed noise and also its intrinsic ability in estimating closely spaced frequency components. The proposed approach is found to be inherently simple, reliable and consistent in performance as well as characterised by low computational burden. Some numerical applications validate the methodology and assess its own performance on synthetic signals, near real‐life signals acquired by IEEE test networks and on a real measured signal from a wide‐area monitoring system currently in operation.

These lecture notes provide a comprehensive self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics computational mathematics and engineering sciences including data assimilation signal processing combinatorial optimization compressed sensing econometrics and mathematical finance among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics non-commutative algebra geometry multivariate functional and spectral analysis and of course statistics and probability theory. As a result most of the classical topics in random matrix theory are technical and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specificcontext of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied along with new spectral and trace concentration-type inequalities.