Dynamics of dislocation densities in a bounded channel. Part II: existence of weak solutions to a singular Hamilton-Jacobi/parabolic strongly coupled system

We study a strongly coupled system consisting of a parabolic equation and a singular Hamilton-Jacobi equation in one space dimension. This system describes the dynamics of dislocation densities in a material submitted to an exterior applied stress. The equations are written on a bounded interval with Dirichlet boundary conditions and require special attention to the boundary. We prove a result of global existence of a solution. The method of the proof consists in considering first a parabolic regularization of the full system, and then passing to the limit. We show some uniform bounds on this solution which uses in particular an entropy estimate for the densities

Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system

We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equations is written on a bounded interval with Dirichlet conditions and requires a special attention to the boundary. The proof of existence and uniqueness is done under the use of two main tools: a certain comparison principle on the gradient of the solution, and a parabolic Kozono-Taniuchi inequalityComment: 36 page

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time $T$ if and only if its energy is negative at some time before $T$. The proof of this result is based on a Gamma-convergence technique

Stability chart of parametric vibrating systems, using energy method

Based on the integral of energy and numerical integration, we introduce, develop, and apply a general algorithm to predict parameters of a parametric equation to produce a periodic response. Using the new method, called energy-rate, we are able to 3nd not only stability chart of a parametric equation which indicates the boundaries of stable and unstable regions, but also periodic responses that are embedded in stable or unstable regions. There are three main important advantages in energy-rate method. It can be applied not only to linear but also to non-linear parametric equations; most of the perturbation methods cannot. It can be applied to large values of parameters; most of the perturbation methods cannot. Depending on the accuracy of numerical integration method, it can also 3nd the value of parameters for a periodic response more accurate than classical methods, no matter if the periodic response is on the boundary of stability and instability or it is a periodic response within the stable or unstable region

Mathematical theory of autodriver for autonomous vehicles

Introducing an independent four-wheel-steering (4WS) system, we are able to design an autodriver to keep an autonomous vehicle on a given road. The kinematic condition of steering can be used to set the steer angles such that the kinematic center of rotation be at any desired point. The road and tire characteristics, along with the dynamics of a moving vehicle cause the vehicle to turn about an actual point that is not necessarily at the road curvature center. The position of the dynamic turning center can be controlled by adjusting the steer angles such that it coincides with the road curvature center. Such a vehicle will move on the desired road autonomously

Theory of autodriver for autonomous vehicles

Introducing an independent four-wheel-steering (4WS) system, we are able to design an autodriver to keep an autonomous vehicle on a given road. The kinematic condition of steering can be used to set the steer angles such that the kinematic center of rotation be at any desired point. The road and tire characteristics, along with the dynamics of a moving vehicle cause the vehicle to turn about an actual point that is not necessarily at the road curvature center. The position of the dynamic turning center can be controlled by adjusting the steer angles such that it coincides with the road curvature center. Such a vehicle will move on the desired road autonomously