Infinite-dimensional State Space Research Articles

Infinite-dimensional reservoir computing

Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state–space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality when learning input/output systems in the class of generalized Barron functionals and measuring the error in a mean-squared sense.

Structured population models on Polish spaces: A unified approach including graphs, Riemannian manifolds and measure spaces to describe dynamics of heterogeneous populations

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes appear in a variety of applications, such as crowd dynamics, tissue regeneration, cancer development and coagulation-fragmentation processes. The diverse applications pose a common challenge to mathematicians due to the multiscale nature of the structures that underlie the system’s self-organization and control. Similar abstract mathematical problems arise when formulating problems in the language of measure evolution on a multi-faceted state space. Motivated by these observations, we propose a general mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. We exploit the structure of the space of non-negative Radon measures with the dual bounded Lipschitz distance (flat metric), which is a generalization of the Wasserstein distance, capable of addressing non-conservative problems. The formulation of models on general metric spaces allows considering infinite-dimensional state spaces or graphs and coupling discrete and continuous state transitions. This opens up exciting possibilities for modeling single-cell data, crowd dynamics or coagulation-fragmentation processes.

Control barrier functionals: Safety‐critical control for time delay systems

AbstarctThis work presents a theoretical framework for the safety‐critical control of time delay systems. The theory of control barrier functions, that provides formal safety guarantees for delay‐free systems, is extended to systems with state delay. The notion of control barrier functionals is introduced, to attain formal safety guarantees by enforcing the forward invariance of safe sets defined in the infinite dimensional state space. The proposed framework is able to handle multiple delays and distributed delays both in the dynamics and in the safety condition, and provides an affine constraint on the control input that yields provable safety. This constraint can be incorporated into optimization problems to synthesize pointwise optimal and provable safe controllers. The applicability of the proposed method is demonstrated by numerical simulation examples.

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Truncated Affine Rozansky–Witten Models as Extended TQFTs

We construct extended TQFTs associated to Rozansky–Witten models with target manifolds T^*mathbb {C}^n. The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category mathcal C of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in mathcal C for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them.

A note on the wave equation controlled with a dynamic saturating boundary control

This paper studies the nonlinear systems obtained by considering a wave equation in closed loop with a nonlinear dynamical boundary controller. The controller is subject to a magnitude limitation and modeled by a linear ordinary differential equation with a saturation map in the input. The well-posedness of the obtained infinite-dimensional system is first studied and then two stability results are given. These two stability results apply for two cascade cases and give sufficient conditions for the asymptotic stability of the equilibrium. The well-posedness is proven by using nonlinear semigroups techniques, whereas the global asymptotic stability results are obtained by Lyapunov-based arguments in infinite-dimensional state space.

A new approach to modeling and simulation of the nonlinear, fractional behavior of Li-ion battery cells

Two nonlinear infinite dimensional models of a Lithium-ion battery are introduced. The proposed models interpolate linear fractional models which are obtained from a widely spread equivalent circuit model. The linear models are parameterized by means of so-called dynamic impedance measurements, i.e. impedance measurements in several operating points. In each operating point, the fractional input output behavior is recovered by an infinite dimensional state space description. For numerical implementation the introduced models require for an appropriate approximation. Therefore, a Krylov subspace method and a finite element approach are proposed. The nonlinear lumped parameter models are validated on the basis of experimental data.

$ L^p $-exact controllability of partial differential equations with nonlocal terms

<p style='text-indent:20px;'>The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces, <inline-formula><tex-math id="M3">\begin{document}$ 1<p<\infty $\end{document}</tex-math></inline-formula>. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.</p>

Many-server asymptotics for join-the-shortest-queue: Large deviations and rare events

The join-the-shortest-queue routing policy is studied in an asymptotic regime where the number of processors n scales with the arrival rate. A large deviation principle (LDP) for the occupancy process is established, as n→∞, in a suitable infinite-dimensional path space. Model features that present technical challenges include, Markovian dynamics with discontinuous statistics, a diminishing rate property of the transition probability rates, and an infinite-dimensional state space. The difficulty is in the proof of the Laplace lower bound which requires establishing the uniqueness of solutions of certain infinite-dimensional systems of controlled ordinary differential equations. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of long queues. In particular, denoting by Ejn(T) the event that there is at least one queue with j or more jobs at some time instant over [0,T], we show that, in the critical case, for large n and T, P(Ejn(T))≈exp[−n(j−2)24T].

Distribution-dependent stochastic differential delay equations in finite and infinite dimensions

We prove that distribution-dependent (also called McKean–Vlasov) stochastic delay equations of the form [Formula: see text] have unique (strong) solutions in finite as well as infinite-dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.

Average Cost Optimality in Partially Observed Lost-Sales Inventory Systems

We consider a partially observable inventory system in which the inventory level can only be observed when it reaches zero, the unmet demand is lost, and replenishment orders must be decided so as to minimize the long-run average cost. This problem has an infinite-dimensional state space and is therefore difficult to analyze. We prove the existence of an average-cost optimal policy, using the so-called vanishing discount factor approach. As a main methodological contribution, we provide a way to verify the key condition -- the uniform boundedness of the relative discounted value function -- for the partially observed system. To accomplish that we construct -- for the partially observed system -- a valid policy which in a certain sense copies the actions of another policy for the process with a different initial state. To our best knowledge, this paper is the first dealing with inventory models with partially observable inventory levels under the long-run average cost criterion.

Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design

In this paper, we address the data-driven modeling of a nonlinear dynamical system while incorporating a priori information. The nonlinear system is described using the Koopman operator, which is a linear operator defined on a lifted infinite-dimensional state-space. Assuming that the L2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L2 gain-preserving data-driven modeling, is formulated. Then, its computationally efficient solution method is presented. An application of the modeling method to feedback controller design is also presented. Aiming for robust stabilization using data-driven control under a poor training dataset, we address the following two modeling problems: (1) Forward modeling: the data-driven modeling is applied to the operating data of a plant system to derive the plant model; (2) Backward modeling: L2 gain-preserving data-driven modeling is applied to the same data to derive an inverse model of the plant system. Then, a feedback controller composed of the plant and inverse models is created based on internal model control, and it robustly stabilizes the plant system. A design demonstration of the data-driven controller is provided using a numerical experiment.

Framework for resource quantification in infinite-dimensional general probabilistic theories

Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. Here, we introduce methods for the quantification of resources in general probabilistic theories (GPTs), focusing in particular on the technical issues associated with infinite-dimensional state spaces. We define a universal resource quantifier based on the robustness measure, and show it to admit a direct operational meaning: in any GPT, it quantifies the advantage that a given resource state enables in channel discrimination tasks over all resourceless states. We show that the robustness acts as a faithful and strongly monotonic measure in any resource theory described by a convex and closed set of free states, and can be computed through a convex conic optimization problem. Specializing to continuous-variable quantum mechanics, we obtain additional bounds and relations, allowing an efficient computation of the measure and comparison with other monotones. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. In particular, we establish exact expressions for various classes of states, including Fock states and squeezed states in the resource theory of nonclassicality and general pure states in the resource theory of entanglement, as well as tight bounds applicable in general cases.

Co-Learning of Task and Sensor Placement for Soft Robotics

Unlike rigid robots which operate with compact degrees of freedom, soft robots must reason about an infinite dimensional state space. Mapping this continuum state space presents significant challenges, especially when working with a finite set of discrete sensors. Reconstructing the robot's state from these sparse inputs is challenging, especially since sensor location has a profound downstream impact on the richness of learned models for robotic tasks. In this work, we present a novel representation for co-learning sensor placement and complex tasks. Specifically, we present a neural architecture which processes on-board sensor information to learn a salient and sparse selection of placements for optimal task performance. We evaluate our model and learning algorithm on six soft robot morphologies for various supervised learning tasks, including tactile sensing and proprioception. We also highlight applications to soft robot motion subspace visualization and control. Our method demonstrates superior performance in task learning to algorithmic and human baselines while also learning sensor placements and latent spaces that are semantically meaningful.

On local exponential stability of equilibrium profiles of nonlinear distributed parameter systems

Local exponential (exp.) stability of nonlinear distributed parameter, i.e. infinite-dimensional state space, systems is considered. A weakened concept of Fréchet differentiability (( Y,X )-Fréchet differentiability) for nonlinear operators defined on Banach spaces is proposed, including the introduction of an alternative space (Y) in the analysis. This allows more freedom in the manipulation of norm-inequalities leading to adapted Fréchet differentiability conditions that are easier to check. Then, provided that the nonlinear semigroup generated by the nonlinear dynamics is Fréchet-differentiable in the new sense, appropriate local exp. stability of the equilibria for the nonlinear system is established. In particular, the nonlinear semigroup has to be Fréchet differentiable on Y and (Y,X)-Fréchet differentiable in order to go back to the original state space X. This approach may be called ”perturbation-based” since exp. stability is also deduced from exp. stability of a linearized version of the nonlinear semigroup. Under adapted Fréchet differentiability assumptions, the main result establishes that local exp. stability of an equilibrium for the nonlinear system is guaranteed as long as the exp. stability holds for the linearized semigroup. The same conclusion holds regarding instability. The theoretical results are illustrated on a convection-diffusion-reaction system.

A natural extension of Markov processes and applications to singular SDEs

On établit une méthode générale pour élargir l’espace d’états d’un processus de Markov, de telle façon que l’ensemble des points ajoutés est polaire. Cette extension est une amélioration de l’extension triviale, dans quel cas le processus est bloqué dans les points ajoutés, et elle produit une technique nouvelle de construction des solutions étendues pour des ED(P) stochastiques, à partir de tous les points de départ, telle qu’elles soient des solutions au moins après tout moment de temps strictement positif. Concrètement, on adopte cette stratégie pour étudier des ED stochastiques avec des coefficients singuliers, sur un espace d’états de dimension infinie (par example des EDP stochastiques de type evolution), pour lesquelles on rencontre des situations où tous les points de l’espace ne sont pas des conditions initiales autorisées. La même chose peut se passer dans la construction des solutions pour des problèmes de martingales ou pour des processus de Markov à partir de formes de Dirichlet (généralisées), pour lesquelles notre nouvelle technique s’applique aussi.

Robust Encoding of a Qubit in a Molecule

We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert state space of rotational states of a rigid body. These codes, which protect against both drift in the body's orientation and small changes in its angular momentum, may be well suited for robust storage and coherent processing of quantum information using rotational states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are compatible with rotational states of diatomic molecules, as well as nuclear states of molecules and atoms. We also describe codes associated with general nonabelian compact Lie groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

Nonlinear phase-amplitude reduction of delay-induced oscillations

Spontaneous oscillations induced by time delays are observed in many real-world systems. Phase reduction theory for limit-cycle oscillators described by delay-differential equations (DDEs) has been developed to analyze their synchronization properties, but it is applicable only when the perturbation applied to the oscillator is sufficiently weak. In this study, we formulate a nonlinear phase-amplitude reduction theory for limit-cycle oscillators described by DDEs on the basis of the Floquet theorem, which is applicable when the oscillator is subjected to perturbations of moderate intensity. We propose a numerical method to evaluate the leading Floquet eigenvalues, eigenfunctions, and adjoint eigenfunctions necessary for the reduction and derive a set of low-dimensional nonlinear phase-amplitude equations approximately describing the oscillator dynamics. By analyzing an analytically tractable oscillator model with a cubic nonlinearity, we show that the asymptotic phase of the oscillator state in an infinite-dimensional state space can be approximately evaluated and non-trivial bistability of the oscillation amplitude caused by moderately strong periodic perturbations can be predicted on the basis of the derived phase-amplitude equations. We further analyze a model of gene-regulatory oscillator and illustrate that the reduced equations can elucidate the mechanism of its complex dynamics under non-weak perturbations, which may be relevant to real physiological phenomena such as circadian rhythm sleep disorders.

Technical Committee on Distributed Parameter Systems [Technical Activities

The IEEE Control Systems Society (CSS) Technical Committee on Distributed Parameter Systems (TC-DPS) aims at fostering exchange, collaboration, and dissemination of new results among researchers from both the control-system theory and control-engineering communities who work on modeling, analysis, and control of distributed parameter systems, that is, systems whose state variables depend on both time and space (then with an infinite-dimensional state space) and described by partial differential equations (PDEs).

Robust Hyperbolic Chaos in Froude Pendulum with Delayed Feedback and Periodic Braking

We indicate a possibility of implementing hyperbolic chaos using a Froude pendulum that is able to produce self-oscillations due to the suspension on a shaft rotating at constant angular velocity, in the presence of time-delay feedback and of periodic braking by the application of additional frictional force. We formulate a mathematical model and carry out its numerical research. In the parameter space we reveal areas of chaotic and regular dynamics using the analysis of Lyapunov exponents and some other diagnostic tools. It is shown that there are regions in the parameter space where the Poincaré stroboscopic map has an attractor, which is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space. We confirm the hyperbolicity of the attractor by numerical calculations including the analysis of angles of intersections of stable and unstable invariant subspaces of vectors of small perturbations for trajectories on the attractor and verify the absence of tangencies between these subspaces.