Stability and Well-posedness of a Nonlinear Railway Track Model

Railway tracks rest on a foundation known for exhibiting nonlinear viscoelastic behavior. Railway track deflections are modeled by a semilinear partial differential equation. This paper studies the stability of solutions to this equation in presence of an input. With the aid of a suitable Lyapunov function, existence and exponential stability of classical solutions is established for certain inputs. The Lyapunov function is further used to find an a-priori estimate of the solutions, and also to study the input-to-state stability (ISS) of mild solutions

Optimal Actuator Design for Nonlinear Systems

For systems modeled by partial differential equations (PDE's), the location and shape of the actuators can be regarded as a design variable and included as part of the controller synthesis procedure. Optimal actuator location is a special case of optimal design. Appropriate actuator location and design can improve performance and significantly reduce the cost of the control in a distributed parameter system. For linear partial differential equations, the existence of an optimal actuator design for a number of cost functions has been established. However, many dynamics are affected by nonlinearities and linearization of the PDE can neglect some important aspects of the model. The existing literature uses the finite-dimensional approximation of nonlinear PDE's to address the optimal actuator design problem. There are new theoretical results on the optimal actuator design of nonlinear PDE's in Banach spaces. This thesis describes new results on optimal actuator design for abstract nonlinear systems on reflexive Banach spaces. Two classes of PDE's have been studied. In the first class, semi-linear systems, a weak continuity assumption on nonlinearities is imposed to establish optimality results. Two examples are provided for this class including nonlinear railway track model and nonlinear wave equation in two space dimensions. The second class is nonlinear parabolic PDE's. For this class, the weak continuity assumption is relaxed at the cost of imposing assumptions on the linear part of the system. The examples provided for this class are Kuramoto-Sivashinsky equation and nonlinear diffusion in two space dimensions. Furthermore, a thorough study of optimal actuator location for nonlinear railway track model was conducted. The study begins with investigating the well-posedness and stability of solutions to this model. It is shown that under certain conditions on inputs, solutions to the railway track model are stable. Further on, using optimization algorithms and numerical schemes, an optimal input and actuator location are computed. Several simulations are run for various physical parameters. The simulations show that the optimal actuator location is not at the center of the track, contrary to a common belief. They also show that an optimally-located actuator significantly improves the performance of the control system

Hybrid control for low-regular nonlinear systems: application to an embedded control for an electric vehicle

This note presents an embedded automatic control strategy for a low consumption vehicle equipped with an "on/off" engine. The main difficulties are the hybrid nature of the dynamics, the non smoothness of the dynamics of each mode, the uncertain environment, the fast changing dynamics, and low cost/ low consumption constraints for the control device. Human drivers of such vehicles frequently use an oscillating strategy, letting the velocity evolve between fixed lower and upper bounds. We present a general justification of this very simple and efficient strategy, that happens to be optimal for autonomous dynamics, robust and easily adaptable for real-time control strategy. Effective implementation in a competition prototype involved in low-consumption races shows that automatic velocity control achieves performances comparable with the results of trained human drivers. Major advantages of automatic control are improved robustness and safety. The total average power consumption for the control device is less than 10 mW

Exponential integrators for second-order in time partial differential equations

Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute efficiently the action of the matrix exponential as well as those of related matrix functions. Various numerical simulations are presented that illustrate this approach.Comment: 19 pages, 10 figure

A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks

In this paper we propose a LWR-like model for traffic flow on networks which allows one to track several groups of drivers, each of them being characterized only by their destination in the network. The path actually followed to reach the destination is not assigned a priori, and can be chosen by the drivers during the journey, taking decisions at junctions. The model is then used to describe three possible behaviors of drivers, associated to three different ways to solve the route choice problem: 1. Drivers ignore the presence of the other vehicles; 2. Drivers react to the current distribution of traffic, but they do not forecast what will happen at later times; 3. Drivers take into account the current and future distribution of vehicles. Notice that, in the latter case, we enter the field of differential games, and, if a solution exists, it likely represents a global equilibrium among drivers. Numerical simulations highlight the differences between the three behaviors and suggest the existence of multiple Wardrop equilibria

Optimal Controller and Actuator Design for Nonlinear Parabolic Systems

Many physical systems are modeled by nonlinear parabolic differential equations, such as the Kuramoto-Sivashinsky (KS) equation. In this paper, the existence of a concurrent optimal controller and actuator design is established for semilinear systems. Optimality equations are provided. The results are shown to apply to optimal controller/actuator design for the Kuramoto-Sivashinsky equation and also nonlinear diffusion

A hierarchy of models for type-II superconductors

A hierarchy of models for type-II superconductors is presented. Through appropriate asymptotic limits we pass from the mesoscopic Ginzburg-Landau model to the London model with isolated superconducting vortices as line singularities, to vortex-density models, and finally to macroscopic critical-state models