Velocity Tracking Problem Research Articles

A Discontinuous Galerkin Time-Stepping Scheme for the Velocity Tracking Problem

The velocity tracking problem for the evolutionary Navier--Stokes equations in two dimensions is studied. The controls are of distributed type and are submitted to bound constraints. First and second order necessary and sufficient conditions are proved. A fully discrete scheme based on the discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, $\tau$ and $h$, respectively, satisfy $\tau \leq Ch^2$, then $L^2$ error estimates of order $O(h)$ are proved for the difference between the locally optimal controls and their discrete approximations.

Error Estimates for the Velocity Tracking Problem for Navier–Stokes Flows Based on the Artificial Compressibility Formulation

ABSTRACT The velocity tracking problem for Navier–Stokes flows is studied based on the artificial compressibility formulation. Several results concerning the analysis and convergence as ϵ → 0 are presented. Symmetric error estimates for semidiscrete finite element approximations for the artificially compressible optimality system are derived.

Fully discrete approximation of a velocity and temperature tracking problem for the Boussinesq equations with piecewise distributed controls

AbstractIn this article, we study the dynamics of a fully discrete piecewise optimal distributed control problem for the Boussinesq equations which models a velocity tracking over time coupled to thermal dynamics. We prove that the rates of velocity and‐ temperature tracking are exponential. We also give some computational results which reinforces the theoretical results derived. Copyright © 2005 John Wiley & Sons, Ltd.

Analysis and finite element error estimates for the velocity tracking problem for Stokes flowsviaa penalized formulation

A distributed optimal control problem for evolutionary Stokes flows is studied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the corresponding optimality system are derived based on estimates for the penalized Stokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, the convergence of the solutions of the penalized optimality systems as ε → 0 is examined.

Analysis and Approximation of the Velocity Tracking Problem for Navier--Stokes Flows with Distributed Control

We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier--Stokes flow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.

Control of Unicycle-Type Robots in the Presence of Sliding Effects with Only Absolute Longitudinal and Yaw Velocities Measurements

In this paper we are concerned with the control of unicycle-type wheeled mobile robots in presence of sliding effects measurements. The slipping and skidding effects are known to be very noisy so we want to elaborate a control law which solves the velocity tracking problem by using only absolute longitudinal and yaw velocities. First, we choose to simplify the problem by taking into account only the more important component of the slip with respect to the trajectory. In a turn, we modelize the skidding effects, however we introduce the longitudinal slip in a straight line. Then, we got our inspiration from the approach used in these particular cases to elaborate a global control law which makes the lateral, longitudinal and yaw velocities tend to zero. This control law does not need sliding effects measurements but depends on the slip and cornering stiffness coefficients.

The Velocity Tracking Problem for Navier--Stokes Flows with Bounded Distributed Controls

We present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for Navier--Stokes flows in bounded two-dimensional domains with bounded distributed controls. We study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined. We also define and analyze semidiscrete-in-time and fully space-time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system. The results of some computational experiments are provided.

Dynamics and Approximations of a Velocity Tracking Problem for the Navier--Stokes Flows with Piecewise Distributed Controls

We study the dynamics of a piecewise (in time) distributed optimal control problem for the Navier--Stokes equations which models velocity tracking over time. We also study the dynamics of semidiscrete and fully discrete approximations of this velocity tracking problem. We prove that the rates of velocity tracking are exponential. Some computational results are presented, which reinforces the theoretical results derived.

Distribution of pressure and temperature in a highly-loaded, lubricated contact : D. Dowson and M. D. Longfield, University of Leeds

In this paper we are concerned with the control of unicycle-type wheeled mobile robots in presence of sliding effects measurements. The slipping and skidding effects are known to be very noisy so we want to elaborate a control law which solves the velocity tracking problem by using only absolute longitudinal and yaw velocities.First, we choose to simplify the problem by taking into account only the more important component of the slip with respect to the trajectory. In a turn, we modelize the skidding effects, however we introduce the longitudinal slip in a straight line. Then, we got our inspiration from the approach used in these particular cases to elaborate a global control law which makes the lateral, longitudinal and yaw velocities tend to zero. This control law does not need sliding effects measurements but depends on the slip and cornering stiffness coefficients.