Control and stabilization of waves on 1-d networks

We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. Dáger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d'Alembert's formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems

A New Approach for the Stability Analysis of Wave Networks

We introduce a new approach to investigate the stability of controlled tree-shaped wave networks and subtrees of complex wave networks. It is motivated by regarding the network as branching out from a single edge. We present the recursive relations of the Laplacian transforms of adjacent edges of the system in its branching order, which form the characteristic equation. In the stability analysis, we estimate the infimums of the recursive expressions in the inverse order based on the spectral analysis. It is a feasible way to check whether the system is exponentially stable under any control strategy or parameter choice. As an application we design the control law and study the stability of a 12-edge tree-shaped wave network

Spectral analysis and Riesz basis property for vibrating systems with damping

In this thesis, we study one-dimensional wave and Euler-Bernoulli beam equations with Kelvin-Voigt damping, and one-dimensional wave equation with Boltzmann damping. The spectral property of equations with clamped boundary conditions and internal Kelvin-Voigt damping are considered. Under some assumptions on the coe±cients, it is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of ¯nite algebraic multiplicity, and the continuous spectrum that is identical to the essential spec- trum is an interval on the left real axis. The asymptotic behavior of eigenvalues is also presented. Two di®erent Boltzmann integrals that represent the memory of materials are consid- ered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the in¯nity, the spectrum of system contains a left half complex plane, which is sharp contrast to most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the later case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: There is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded

Basis expansions in applied mathematics

Basis expansions are an extremely useful tool in applied mathematics. By using them, we can express a function representing a physical quantity as a linear combination of simpler ``modules'' with well-known properties. They are particularly useful for the applications described in this thesis. Perhaps the best known expansion of this type is the Fourier series of a periodic function, as decomposition into the infinite sum of simple sinusoidal and cosinusoidal elements, originally proposed by Fourier to study heat transfer. This dissertation employs some mathematical tools on problems taken from various areas of Engineering, exploiting their expansion properties: 1) Non-integer bases, which are applied to mathematical models in Robotics (Chapter 2). In this Chapter we study, in particular, a model for snake-like robots based on the Fibonacci sequence. It includes an investigation of the reachableworkspace, a more general analysis of the control system underlying the model, its reachability and local controllability properties. 2) Orthonormal bases, Riesz bases: exponential and cardinal Riesz basis with perturbations (Chapter 3). In this Chapter we obtain also a stability result for cardinal Riesz basis in the case of complex perturbations of the integers. We also consider a mathematical model for energy of the signal at the output of an ideal DAC, in presence of sampling clock jitter. When sampling clock jitter occurs, the energy of the signal at the output of ideal DAC does not satisfies a Parseval identity. Nevertheless, an estimation of the signal energy is here shown by a direct method involving cardinal series. 3) Orthogonal polynomials (Chapter 4). In this Chapter we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev polynomials of the first and second kind. We discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits 0 and 1 to + and - signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which take the form of nested square roots of 2. These zeros are used to obtain two new formulas for Pi

Spectral Analysis of Complex Dynamical Systems

The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that lateral vibrations of such strings can be decoupled from their in-plane vibrations. We then study the eigenvalue problem originating from the lateral vibrations. We show that, unlike the well-known one string vibrations case, the eigenvalues in a multi-string vibrating system do not have to be simple. Moreover we prove that the multiplicities of the eigenvalues depend on the symmetry of the model and on the total number of the connected strings [50]. We also apply Nevanlinna functions theory to characterize the spectra and to solve the inverse problem for a discrete multi-string system in a more general setting than it was done in [71],[73], [22], [69]-[72]. We also represent multi-string vibrating systems using a coupling of non-densely defined symmetric operators acting in the infinite dimensional Hilbert space. This coupling is defined by a special set of boundary operators acting in finite dimensional Krein space (the space with indefinite inner product). The main results of this research are published in [50]. The Hypothalamic Pituitary Adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body’s cortisol level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor but does not take into account the system response delay, we first perform rigorous stability analysis of all multi-parametric steady states and secondly, by construction of a Lyapunov functional, we prove nonlinear asymptotic stability for some of multi-parametric steady states. We then take into account the additional effects of the time delay parameter on the stability of the HPA axis system. Finally we prove the existence of periodic solutions for the HPA axis system. The main results of this research are published in [51]

Advances in Computer Science and Engineering

The book Advances in Computer Science and Engineering constitutes the revised selection of 23 chapters written by scientists and researchers from all over the world. The chapters cover topics in the scientific fields of Applied Computing Techniques, Innovations in Mechanical Engineering, Electrical Engineering and Applications and Advances in Applied Modeling

Quantum brane cosmology

This thesis deals with the interaction of quantum mechanical models and cosmologies based on brane universes, an area of active theoretical speculation over the last five years.For convenience, the material has been split into two parts. Part 1 deals with a selection of background topics which are necessary and relevant to the original research. This research is presented in Part 2. In addition, some auxiliary topics, both more elementary and more advanced, are described in the appendices. The selection of background topics has been influenced by the various techniques, physical theories and mathematical technologies which play a major role in the work presented in Part 2. Although the exposition is ad hoc, an attempt has been made to systematically develop portions where the technique (or use of it) may be unfamiliar.A fairly complete treatment of the necessary mathematical scaffolding is supplied. Although important, this material is familiar or strongly mathematical, and is deferred to the appendices. This includes an elementary survey of functional analysis in Appendix A, sufficient to support a discussion of the path integral. The path integral formalism is used extensively throughout this thesis, and, where available, constitutes our preferred representation of quantum mechanics. The discussion is limited to the relevant portions of the theory: functions in Banacli spaces, and the Sturm-Liouville basis (technology which appears many times in Part 2); direct evaluation of Gaussian functional integrals, ubiquitous in field theory calculations; and ((-function regularization of the operator determinants to which such Gaussian integrals give rise, which has a direct application in Chapter 9. In Appendix B we describe the necessary framework of differential geometry which supports general relativity, and low-energy discussions of string theory. All calculations in metric gravity are based on differential geometry, together with a good proportion of the technology which buttresses quantum field theory on curved space time, string theory, and some more advanced representations of quantum mechanics (see below). All of this is used extensively throughout both parts of the thesis. We include some more advanced topological technology which supports the discussion of string compactification. General results from compactification theory, when appropriately interpreted in the brane context, contribute important stability results for zero-modes of the Kaluza—Klein fields, and provide a natural home for the spectral KK technology used (in one form or another) throughout Part 2, but most especially in Chapter 7 and Chapter 8. Einstein gravity and Yang-Mills theory are set in context as examples of connexions on fibre bundles

Modern applications of machine learning in quantum sciences

In these Lecture Notes, we provide a comprehensive introduction to the most recent advances in the application of machine learning methods in quantum sciences. We cover the use of deep learning and kernel methods in supervised, unsupervised, and reinforcement learning algorithms for phase classification, representation of many-body quantum states, quantum feedback control, and quantum circuits optimization. Moreover, we introduce and discuss more specialized topics such as differentiable programming, generative models, statistical approach to machine learning, and quantum machine learning

Modern applications of machine learning in quantum sciences

In these Lecture Notes, we provide a comprehensive introduction to the most recent advances in the application of machine learning methods in quantum sciences. We cover the use of deep learning and kernel methods in supervised, unsupervised, and reinforcement learning algorithms for phase classification, representation of many-body quantum states, quantum feedback control, and quantum circuits optimization. Moreover, we introduce and discuss more specialized topics such as differentiable programming, generative models, statistical approach to machine learning, and quantum machine learning.Comment: 268 pages, 87 figures. Comments and feedback are very welcome. Figures and tex files are available at https://github.com/Shmoo137/Lecture-Note