Infinite-dimensional Systems Research Articles

Path integrals, complex probabilities and the discrete Weyl representation

A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of the Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. The method is illustrated by applying it to scattering theory and quantum field theory. The implications of these applications for quantum computing is discussed.

Adaptive identification of linear infinite-dimensional systems

We propose an adaptive algorithm for identifying the unknown parameter (possibly vector-valued) in a linear stable single-input single-output infinite-dimensional system. We assume that the transfer function of the infinite-dimensional system can be expressed as a ratio of two infinite series in s (the Laplace variable). We also assume that certain identifiability conditions, which include a persistency of excitation condition, hold. For a fixed integer n, we propose an adaptive update law (a differential equation in time) driven by real-time input-output data for estimating the first n + 1 coefficients in the numerator and the denominator of the transfer function. We show that the estimates for the transfer function coefficients generated by the update law are close to the true values at large times provided n is sufficiently large (the estimates converge to the true values as time and n tend to infinity). The unknown parameter can be reconstructed using the transfer function coefficient estimates obtained with n large and the algebraic expressions relating the transfer function coefficients to the unknown parameter. We also provide a numerical scheme for verifying the identifiability conditions and for choosing n sufficiently large so that the value of the reconstructed parameter is close to the true value. The class of systems to which our approach is applicable includes many partial differential equations with constant/spatially-varying coefficients and distributed/boundary input and output. We illustrate the efficacy of our approach using three examples: a delay system with four unknown scalars, a 1D heat equation with two unknown scalars and a 1D wave equation with an unknown linearly-varying (in space) coefficient.

Modeling spatiotemporal temperature dynamics of large-format power batteries: A multi-source information fusion approach

Uneven temperature distributions can significantly impact battery performance and cycle life. Industrial applications, in particular, where unknown disturbances and limited sensors exist, pose significant challenges for accurately estimating the battery temperature field. This work proposes a multi-source information fusion framework for capturing the spatiotemporal temperature dynamics of pouch-type Lithium-ion batteries. First, we develop a system model of the battery thermal process based on physical insights, considering unknown parameter deviations and unmodeled dynamics. Next, we use the Galerkin-spectral method to expand the spatiotemporal variable and reduce the original infinite-dimensional system to a low-order model containing the most important system modes. The unknown component of the model is then entirely stripped and integrated into a nonlinear term. To close the reality gap of the physics-based model, we subsequently develop an error compensation model that learns the dynamic behavior from sparse observations. Ultimately, our framework design yields a fusion-driven model for reliable temperature field prediction. Simulations and experimental data validate the superior performance and generalization capability of the proposed method.

Galerkin approximation for [formula omitted]-control of the stable parabolic system under Dirichlet boundary control

In this paper, we explore state feedback control for the H∞ disturbance-attenuation problem in stable parabolic systems with in-domain distributed disturbances under Dirichlet boundary control. Calculating the state feedback control involves solving an operator algebraic Riccati equation, which poses challenges in finding an analytic solution. A practical approach is to seek an approximate solution via finite-dimensional approximation. Specifically, we employ the Galerkin approximation, which generates a sequence of finite-dimensional systems that approximate the original infinite-dimensional system. All corresponding finite-dimensional disturbance-attenuation problems are solvable, and it is demonstrated that the sequence of solutions to the associated finite-dimensional algebraic Riccati equations converges in norm to the solution of the infinite-dimensional operator algebraic Riccati equation. Furthermore, the state feedback controls derived from the finite-dimensional algebraic Riccati equations are proven to be γ-admissible controls for the original system.

LOCC convertibility of entangled states in infinite-dimensional systems

We advance on the conversion of bipartite quantum states via local operations and classical communication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.

О потенциальности, дискретизации и интегральных инвариантах бесконечномерных систем Биркгофа

О потенциальности, дискретизации и интегральных инвариантах бесконечномерных систем Биркгофа

Closed-loop Koopman operator approximation

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop (CL) system, to simultaneously identify the CL and plant systems. The advantages of the proposed CL Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

Large deviation principle for stochastic FitzHugh–Nagumo lattice systems

This paper is devoted to the large deviation principle of stochastic FitzHugh–Nagumo lattice system, where the nonlinear drift and diffusion terms are locally Lipschitz continuous. Based on the well-posedness and the convergence of the solutions of the controlled system associated with the considered system, by the weak convergence method, we establish the large deviation principle in C([0,T],l2×l2) for such infinite dimensional system.

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

Stark Many-Body Localization in Interacting Infinite Dimensional Systems.

We study bulk particle transport in a Fermi-Hubbard model on an infinite-dimensional Bethe lattice, driven by a constant electric field. Previous numerical studies showed that one dimensional analogs of this system exhibit a breakdown of diffusion due to Stark many-body localization at least up to time that scales exponentially with the system size. Here, we consider systems initially in a spin density wave state using a combination of numerically exact and approximate techniques. We show that for sufficiently weak electric fields, the wave's momentum component decays exponentially with time in a way consistent with normal diffusion. By studying different wavelengths, we extract the dynamical exponent and the generalized diffusion coefficient at each field strength. Interestingly, we find a nonmonotonic dependence of the dynamical exponent on the electric field. As the field increases toward a critical value proportional to the Hubbard interaction strength, transport slows down, becoming subdiffusive. At large interaction strengths, however, transport speeds up again with increasing field, exhibiting superdiffusive characteristics when the electric field is comparable to the interaction strength. Eventually, at the large field limit, localization occurs and the current through the system is suppressed.

Seizure detection via deterministic learning feature extraction

Epileptic seizures have a significant impact on the well-being of a large number of individuals worldwide. Utilizing electroencephalographic (EEG) signals for automatic seizure detection proves to be a valuable solution. However, dealing with raw EEG signals is inherently complex, necessitating a preliminary step of feature extraction prior to detection. Traditional feature extraction methods often amalgamate various types of features for seizure detection, as each type typically captures specific properties. In contrast, this paper focuses on detecting seizures by analyzing the system dynamics. The proposed Deterministic Learning Feature Extraction (DLFE) method extracts a single type of nonlinear dynamical feature rooted in the EEG system dynamics. DLFE employs deterministic learning to discern the inherent system dynamics of the EEG under both seizure and normal conditions. Through the feature extraction process, the infinite-dimensional system dynamics are transformed into feature vectors, exhibiting distinct distributions in seizure and normal states. This disparity can be effectively utilized for classification using standard classifiers. The performance of the proposed seizure detection method was assessed using the CHB-MIT and Bonn datasets. The average classification accuracy was found to be 98.63% with a specificity of 99.19% and a sensitivity of 98.06% on CHB-MIT dataset. Compared with the latest similar methods, the accuracy, specificity and sensitivity are improved by 0.31%, 0.21% and 0.05% respectively. Moreover, the performance was achieved with the short-time interval EEG signals within a few channels. The average classification accuracy was found to be 99.90% with a 0.22% improvement on Bonn dataset, which indicates the good generalization performance.

State Dependent Delay Maps: Numerical Algorithms and Dynamics of Projections

This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a function of the state variable. A delay map with constant delays can be rewritten explicitly as a discrete time dynamical system on an appropriate function space, and a delay map with small state dependent terms can be viewed as a “non-autonomous” perturbation. We develop a fixed point formulation for the Cauchy problem of such perturbations, and under appropriate assumptions obtain the existence of forward iterates of the map. The proof is constructive and leads to numerical procedures which we implement for illustrative examples, including the cubic Ikeda and Mackey-Glass systems with constant and state-dependent delays. After proving a local convergence result for the method, we study more qualitative/global convergence issues using data analytic tools for time series analysis (dimension and topological measures derived from persistent homology). Using these tools we quantify the convergence of the dynamics in the finite dimensional projections to the dynamics of the infinite dimensional system.

To geometric aspects of in nite-dimensional dynamical systems

The main goal of the work is to construct analogues of Christoffel symbols for infinitedimensional systems and on this basis to obtain geodesic equations for such systems. These analogies are of particular interest in terms of identifying the relationship between the dynamics of systems with an infinite number of degrees of freedom and Riemannian geometry, as well as geometry defined by the pseudo-Riemannian metric.

The subadditivity of quantum entropy in Gaussian quantum systems

The entropy of a quantum state measures information or uncertainty contained in the quantum system and plays a crucial role in quantum information theory. Many entropic properties, such as the subadditivity, have been studied for the discrete system with a finite-dimensional state space, while much less is explored in the infinite-dimensional system. In this work, we study the entropic properties of the continuous system that models a collection of Gaussian modes or fields. In particular, we identify the parameter range for the Rényi, Tsallis, and Unified entropies of Gaussian states that they are additive, subadditive, and strong subadditive. We show that the Rényi entropies are subadditive, further giving rise to the subadditivity of the Tsallis and Unified entropies in certain parameter range. We also find that the Tsallis and Unified entropies are not strong subadditive, while the Rényi entropies with the order ranging from 1 to 2 are conjectured to admit the strong subadditivity. Our results are useful in studying quantum correlations and channel capacities in Gaussian systems.

Stability Radii of Infinite-Dimensional Discrete-Time Systems Discomfited by Stochastic Perturbations

Stability Radii of Infinite-Dimensional Discrete-Time Systems Discomfited by Stochastic Perturbations

KAM theorem for degenerate infinite-dimensional reversible systems

In this article, we establish a Kolmogorov-Arnold-Moser (KAM) theorem for degenerate infinite-dimensional reversible systems under a non-degenerate condition of Russmann type. This theorem broadens the scope of applicability of degenerate KAM theory, previously confined to Hamiltonian systems, by incorporating infinite-dimensional reversible systems. Using this theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of non-Hamiltonian but reversible beam equations with non-linearities in derivatives. For more information see https://ejde.math.txstate.edu/Volumes/2024/02/abstr.html

On BIBO Stability of Infinite-Dimensional Linear State-Space Systems

On BIBO Stability of Infinite-Dimensional Linear State-Space Systems

Reduced order in domain control of distributed parameter port-Hamiltonian systems via energy shaping

An in-domain finite dimensional controller for a class of distributed parameter systems on a one-dimensional spatial domain formulated under the port-Hamiltonian framework is presented. Based on (Trenchant et al. 2017) where positive feedback and a late lumping approach is used, we extend the Control by Interconnection method and propose a new energy shaping methodology with an early lumping approach on the distributed spatial domain of the system. Our two main control objectives are to stabilize the closed-loop system, as well as to improve the closed-loop dynamic performances. With the early lumping approach, we investigate two cases of the controller design, the ideal case where each distributed controller acts independently on the spatial domain (fully-actuated), and the more realistic case where the control action is piecewise constant over certain intervals (under-actuated). We then analyze the asymptotic stability of the closed-loop system when the infinite dimensional plant system is connected with the finite dimensional controller. Furthermore we provide simulation results comparing the performance of the fully-actuated case and the under-actuated case with an example of an elastic vibrating string.

Representing the dissipation of infinite-dimensional linear port-Hamiltonian systems

It is well known that linear and non-linear dissipative port-Hamiltonian systems in finite dimensions admit an energy balance, relating the energy increase in the system with the supplied energy and the dissipated energy. The integrand in the dissipation term is then a function of the state variable. In this note, we answer the question of when this is possible for linear port-Hamiltonian systems in infinite dimensions.

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 non-quadratic 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.