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

On the monotone and primal-dual active set schemes for ℓp\ell^p-type problems, p∈(0,1]p \in (0,1]

Nonsmooth nonconvex optimization problems involving the $\ell^p$ quasi-norm, $p \in (0, 1]$, of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction

Parameter estimation for the Euler-Bernoulli-beam

An approximation involving cubic spline functions for parameter estimation problems in the Euler-Bernoulli-beam equation (phrased as an optimization problem with respect to the parameters) is described and convergence is proved. The resulting algorithm was implemented and several of the test examples are documented. It is observed that the use of penalty terms in the cost functional can improve the rate of convergence

Optimal actuator design based on shape calculus

An approach to optimal actuator design based on shape and topology optimisation techniques is presented. For linear diffusion equations, two scenarios are considered. For the first one, best actuators are determined depending on a given initial condition. In the second scenario, optimal actuators are determined based on all initial conditions not exceeding a chosen norm. Shape and topological sensitivities of these cost functionals are determined. A numerical algorithm for optimal actuator design based on the sensitivities and a level-set method is presented. Numerical results support the proposed methodology.Comment: 41 pages, several figure

Infinite horizon sparse optimal control

A class of infinite horizon optimal control problems involving $L^p$-type cost functionals with $0<p\leq 1$ is discussed. The existence of optimal controls is studied for both the convex case with $p=1$ and the nonconvex case with $0<p<1$, and the sparsity structure of the optimal controls promoted by the $L^p$-type penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers

A convex analysis approach to optimal controls with switching structure for partial differential equations

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach