Mini-Workshop: Wellposedness and Controllability of Evolution Equations

This mini-workshop brought together mathematicians engaged in partial differential equations, operator theory, functional analysis and harmonic analysis in order to address a number of current problems in the wellposedness and controllability of infinite-dimensional systems

Modelling and control of coupled infinite-dimensional systems

First, we consider two classes of coupled systems consisting of an infinite-dimensional part [sigma]d and a finite-dimensional part [sigma]f connected in feedback. In the first class of coupled systems, we assume that the feedthrough matrix of [sigma]f is 0 and that [sigma]d is such that it becomes well-posed and strictly proper when connected in cascade with an integrator. Under several assumptions, we derive well-posedness, regularity and exact (or approximate) controllability results for such systems on a subspace of the natural product state space. In the second class of coupled systems, [sigma]f has an invertible first component in its feedthrough matrix while [sigma]d is well-posed and strictly proper. Under similar assumptions, we obtain well-posedness, regularity and exact (or approximate) controllability results as well as exact (or approximate) observability results for this class of coupled systems on the natural state space. Second, we investigate the exact controllability of the SCOLE (NASA Spacecraft Control Laboratory Experiment) model. Using our theory for the first class of coupled systems, we show that the uniform SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space. Third, we investigate the suppression of the vibrations of a wind turbine tower using colocated feedback to achieve strong stability. We decompose the system into a non-uniform SCOLE model describing the vibrations in the plane of the turbine axis, and another model consisting of a non-uniform SCOLE system coupled with a two-mass drive-train model (with gearbox), in the plane of the turbine blades. We show the strong stabilizability of the first tower model by colocated static output feedback. We also prove the generic exact controllability of the second tower model on a smoother state space using our theory for the second class of coupled systems, and show its generic strong stabilizability on the energy state space by colocated feedback

Control and inverse problems for the wave equation on metric graphs

Dissertation (Ph.D.) University of Alaska Fairbanks, 2022This thesis focuses on control and inverse problems for the wave equation on finite metric graphs. The first part deals with the control problem for the wave equation on tree graphs. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The second part deals with the inverse problem for the wave equation on tree graphs. We describe the dynamical Leaf Peeling (LP) method. The main step of the method is recalculating the response operator from the original tree to a peeled tree. The LP method allows us to recover the connectivity, potential function on a tree graph and the lengths of its edges from the response operator given on a finite time interval. In the third part we consider the control problem for the wave equation on graphs with cycles. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if Neumann controllers are placed at the active vertices and Dirichlet controllers are placed at the active edges. The control time for this construction is determined by the chosen orientation and path decomposition of the graph. We indicate the optimal time that guarantees the exact controllability for all systems of a described class on a given graph. While the choice of the active vertices and edges is not unique, we find the minimum number of controllers to guarantee the exact controllability as a graph invariant.National Science Foundation Graduate Research Fellowship Grant No. 1242789Chapter 1: General Introduction. Chapter 2: Control problems for the wave equation on metric tree graphs -- 2.1. Introduction -- 2.2. Preliminaries -- 2.3. The forward and control problems for the wave equation on a finite length interval -- 2.4. The forward and control problems in a star-shaped neighborhood graph of an internal vertex -- 2.5. Solving the forward problem for wave equations on general graphs -- 2.6. Controllability on a tree graph. Chapter 3: Inverse problem for the wave equation on graphs -- 3.1 Introduction --3.2 Preliminaries -- 3.3 The forward problem and the Duhamel's principle -- 3.4 The response function and the inverse problem -- 3.5 Leaf peeling method on a rooted tree. Chapter 4: Control problems for the wave equations on graphs with cycles -- 4.1. Introduction -- 4.2. Preliminaries -- 4.2.1. Metric graphs and Hilbert spaces on graphs -- 4.2.2. Observation and control problems of the wave equation -- 4.2.3. Directed acyclic graphs and linear ordering of vertices -- 4.2.4. The forward problem on an interval -- 4.2.5. Solution to the forward problem on a general graph -- 4.3. The forward and control problems on a DAG with controllers placed on a single-track active set -- V4.3.1. The tangle-free path union and single-track active set of a DAG -- 4.3.2. The forward problem when {I∗, j∗} is a ST active set -- 4.3.3. Shape and velocity controllability on an interval -- 4.3.4. Shape and velocity controllability on graphs -- 4.3.5. Exact controllability on graphs -- 4.3.6. Connectivity of the graph -- 4.3.7. The number of controllers -- 4.4. Appendix. Chapter 5: Conclusions -- References

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

Joint state and parameter estimation for distributed mechanical systems

We present a novel strategy to perform estimation for a dynamical mechanical system in standard operating conditions, namely, without ad hoc experimental testing. We adopt a sequential approach, and the joint state-parameter estimation procedure is based on a state estimator inspired from collocated feedback control. This type of state estimator is chosen due to its particular effectiveness and robustness, but the methodology proposed to adequately extend state estimation to joint state-parameter estimation is general, and - indeed -applicable with any other choice of state feedback observer. The convergence of the resulting joint estimator is mathematically established. In addition, we demonstrate its effectiveness with a biomechanical test problem defined to feature the same essential characteristics as a heart model, in which we identify localized contractility and stiffness parameters using measurements of a type that is available in medical imaging

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe