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

Finite-time stabilization of a network of strings

We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node

Wind-Driven Gas Networks and Star Formation in Galaxies: Reaction-Advection Hydrodynamic Simulations

The effects of wind-driven star formation feedback on the spatio-temporal organization of stars and gas in galaxies is studied using two-dimensional intermediate-representational quasi-hydrodynamical simulations. The model retains only a reduced subset of the physics, including mass and momentum conservation, fully nonlinear fluid advection, inelastic macroscopic interactions, threshold star formation, and momentum forcing by winds from young star clusters on the surrounding gas. Expanding shells of swept-up gas evolve through the action of fluid advection to form a ``turbulent'' network of interacting shell fragments whose overall appearance is a web of filaments (in two dimensions). A new star cluster is formed whenever the column density through a filament exceeds a critical threshold based on the gravitational instability criterion for an expanding shell, which then generates a new expanding shell after some time delay. A filament- finding algorithm is developed to locate the potential sites of new star formation. The major result is the dominance of multiple interactions between advectively-distorted shells in controlling the gas and star morphology, gas velocity distribution and mass spectrum of high mass density peaks, and the global star formation history. The gas morphology observations of gas in the LMC and in local molecular clouds. The frequency distribution of present-to-past average global star formation rate, the distribution of gas velocities in filaments (found to be exponential), and the cloud mass spectra (estimated using a structure tree method), are discussed in detail.Comment: 40 pp, 15 eps figs, mnras style, accepted for publication in MNRAS, abstract abridged, revisions in response to referee's comment

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

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Patient-specific modelling of the cerebral circulation for aneurysm risk assessment

Cerebral aneurysms are localised pathological dilatations of cerebral arteries, most commonly found in the circle of Willis. Although not all aneurysms are unstable, the major clinical concern involved is the risk of rupture. High morbidity and mortality rates are associated with the haemorrhage resulting from rupture. New indicators of aneurysm stability are sought, since current indicators based on morphological factors have been shown to be unreliable. Haemodynamical factors are known to be relevant in vascular wall remodelling, and therefore believed to play an important role in aneurysmdevelopment and stability. Studies suggest that intra-aneurysmal wall shear stress and flow patterns, for example, are candidate parameters in aneurysm stability assessment. These factors can be estimated if the 3D patient-specific intra-aneurysmal velocity is known, which can be obtained via a combination of in vivo measurements and computational fluid dynamics models. The main determinants of the velocity field are the vascular geometry and flow through this geometry. Over the last decade the extraction of the vascular geometry has become well established. More recently, there has been a shift of attention towards extracting boundary conditions for the 3D vascular segment of interest. The aim of this research is to improve the reliability of the model-based representation of the velocity field in the aneurysmal sac. To this end, a protocol is proposed such that patient-specific boundary conditions for the 3D segment of interest can be estimated without the need for added invasive procedures. This is facilitated by a 1D wave propagation model based on patient-specific geometry and boundary conditions measured non-invasively in more accessible regions. Such a protocol offers improved statistical reliability owing to the increased number of patients that can participate in studies aiming to identify parameters of interest in aneurysm stability assessment. In chapter 2 the intra-aneurysmal velocity field in an idealised aneurysm model is validated with particle image velocimetry experiments, after which the flow patterns are evaluated using a vortex identification method. Chapter 3 describes a 1D model wave propagation model of the cerebral circulation with a patient-specific vascular geometry. The resulting flow pulses at the boundaries of the 3D segment of interest are compared to those obtained with a patient-generic geometry. The influence of these different boundary conditions on the 3D intra-aneurysmal velocity field is evaluated in chapter 4 by prescribing the end-diastolic flows extracted from the 1D models. In order to measure blood flow with videodensitometric methods, an injection of contrast agent is required. The effect of this injection on the flow of interest is assessed in chapter 5. In chapter 6, pressure measurements in the internal carotid are used to evaluate the variability of pressure waveform and its effect on the boundary conditions for the 1D model. Finally, a protocol for full patient-specific modelling is discussed in chapter 7