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

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

Time-Optimal Adiabatic-Like Expansion of Bose-Einstein Condensates

In this paper we study the fast adiabatic-like expansion of a one-dimensional Bose-Einstein condensate (BEC) confined in a harmonic potential, using the theory of time-optimal control. We find that under reasonable assumptions suggested by the experimental setup, the minimum-time expansion occurs when the frequency of the potential changes in a bang-bang form between the permitted values. We calculate the necessary expansion time and show that it scales logarithmically with large values of the expansion factor. This work is expected to find applications in areas where the efficient manipulations of BEC is of utmost importance. As an example we present the field of atom interferometry with BEC, where the wavelike properties of atoms are used to perform interference experiments that measure with unprecedented precision small shifts induced by phenomena like rotation, acceleration, and gravity gradients.Comment: Submitted to 51st IEEE Conference on Decision and Contro

Total variation regularization of multi-material topology optimization

This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient.