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This paper discusses the boundary stabilization of a beam in free transverse vibration. The dynamics of the beam is presented by a nonlinear partial differential equation (PDE). Based on this model a nonlinear control law is constructed to stabilize the system. The control law is a nonlinear function of the slopes and velocity at the boundary of the beam. The novelty of this article is that it has been possible to exponentially stabilize a free transversely vibrating beam via boundary control without restoring to truncation of the model. This result is achieved while the coupling between longitudinal and transversal displacements has been taken into account.

The Lagrangian equations for distributed-parameter systems based on Hamilton's principle are developed. These equations are subsequently used to derive nonlinear models for beams. The passivity properties of the flexible mechanical systems based on their distributed-parameter models are then investigated and direct output feedback control laws for control purposes are proposed. Finite gain L2 stability and passivity of closed-loop systems are proven. Illustrative cases with simulation of the nonlinear beams and stabilizing feedback control laws are included in the text

Summary With the recent developments in the oil and gas industry, in which the number of satellite fields are increasing, the industry faces major challenges. A mixture of gas and oil is transported in multiphase pipelines along the bottom of the sea from the wells and up through the riser to the oil rig. The development of unstable flow in multiphase pipelines is a major and expensive problem. The irregular flow results in poor oil/water separation, limits the production capacity, and causes flaring. Handling slug flow thus has become very important with regard to increasing the production rate and avoiding possible equipment damages. In this paper, we have developed a model of a physical process that generates slugs. The model has been verified against experimental data. Several control strategies have been tested on the model, and the simulation results are presented. It is shown by simulation of multiphase flow that the unstable slug flow can be stabilized by feedback control. In addition, interesting and important phenomena such as inverse response of the pressure at the top and asymmetric response of the pressure at the bottom have been revealed and explained. Introduction Multiphase pipelines connecting remote wellhead platforms and subsea wells are a common feature of offshore oil production. Recent developments indicate that this trend will be followed in the future. In addition, the feasibility of using long-distance tieback pipelines to connect subsea processing units directly to onshore processing plants makes it likely that these will also be deployed in the future. These trends in offshore oil production face a major challenge. The slug phenomenon in multiphase flow lines with both gas and liquid hydrocarbons has gained increased interest in recent years. Slugging can be characterized as either hydrodynamic or terrain slugging. Hydrodynamic slugs are built in horizontal parts of the pipeline when liquid and gas velocities are different. These slugs are usually short with higher frequency. The inlet separator will in most cases handle these slugs with fewer difficulties, because the amount of liquid is small compared to the volume of the separator. On the contrary, a terrain slug can contain a lot of liquid and represents a great challenge to the downstream processing system. Level and pressure variations in the first-stage separator can be propagated in the downstream process. Variations in the separator level may cause poor separation and occasionally fluid flooding. Oscillations in the separator pressure can result in increased flaring, which is environmentally benign and is also costly because of CO2 taxes. The degree of slugging depends on various factors. The most important are pipeline pressure, pipeline topology, and production rate. Terrain slugging is most likely to occur at low rates, with a low pipeline pressure toward the end of a well's lifetime.