446 research outputs found
An adaptive POD approximation method for the control of advection-diffusion equations
We present an algorithm for the approximation of a finite horizon optimal
control problem for advection-diffusion equations. The method is based on the
coupling between an adaptive POD representation of the solution and a Dynamic
Programming approximation scheme for the corresponding evolutive
Hamilton-Jacobi equation. We discuss several features regarding the adaptivity
of the method, the role of error estimate indicators to choose a time
subdivision of the problem and the computation of the basis functions. Some
test problems are presented to illustrate the method.Comment: 17 pages, 18 figure
Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls
The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on 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, τ and h respectively, satisfy τ ≤ Ch2 , then L2 error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P
Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems
This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop
Time-optimality by distance-optimality for parabolic control systems
The equivalence of time-optimal and distance-optimal control problems is
shown for a class of parabolic control systems. Based on this equivalence, an
approach for the efficient algorithmic solution of time-optimal control
problems is investigated. Numerical examples are provided to illustrate that
the approach works well is practice
Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations
An advantageous feature of piecewise constant policy timestepping for
Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation
schemes, and indeed different meshes, can be used for the resulting linear
equations for different control parameters. Standard convergence analysis
suggests that monotone (i.e., linear) interpolation must be used to transfer
data between meshes. Using the equivalence to a switching system and an
adaptation of the usual arguments based on consistency, stability and
monotonicity, we show that if limited, potentially higher order interpolation
is used for the mesh transfer, convergence is guaranteed. We provide numerical
tests for the mean-variance optimal investment problem and the uncertain
volatility option pricing model, and compare the results to published test
cases
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