Norm and time optimal control problems of stochastic heat equations

This paper investigates the norm and time optimal control problems for stochastic heat equations. We begin by presenting a characterization of the norm optimal control, followed by a discussion of its properties. We then explore the equivalence between the norm optimal control and time optimal control, and subsequently establish the bang-bang property of the time optimal control. These problems, to the best of our knowledge, are among the first to discuss in the stochastic case

Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.Comment: this is a living document - we welcome feedback and discussio

A Multigrid Method for the Efficient Numerical Solution of Optimization Problems Constrained by Partial Differential Equations

We study the minimization of a quadratic functional subject to constraints given by a linear or semilinear elliptic partial differential equation with distributed control. Further, pointwise inequality constraints on the control are accounted for. In the linear-quadratic case, the discretized optimality conditions yield a large, sparse, and indefinite system with saddle point structure. One main contribution of this thesis consists in devising a coupled multigrid solver which avoids full constraint elimination. To this end, we define a smoothing iteration incorporating elements from constraint preconditioning. A local mode analysis shows that for discrete optimality systems, we can expect smoothing rates close to those obtained with respect to the underlying constraint PDE. Our numerical experiments include problems with constraints where standard pointwise smoothing is known to fail for the underlying PDE. In particular, we consider anisotropic diffusion and convection-diffusion problems. The framework of our method allows to include line smoothers or ILU-factorizations, which are suitable for such problems. In all cases, numerical experiments show that convergence rates do not depend on the mesh size of the finest level and discrete optimality systems can be solved with a small multiple of the computational cost which is required to solve the underlying constraint PDE. Employing the full multigrid approach, the computational cost is proportional to the number of unknowns on the finest grid level. We discuss the role of the regularization parameter in the cost functional and show that the convergence rates are robust with respect to both the fine grid mesh size and the regularization parameter under a mild restriction on the next to coarsest mesh size. Incorporating spectral filtering for the reduced Hessian in the control smoothing step allows us to weaken the mesh size restriction. As a result, problems with near-vanishing regularization parameter can be treated efficiently with a negligible amount of additional computational work. For fine discretizations, robust convergence is obtained with rates which are independent of the regularization parameter, the coarsest mesh size, and the number of levels. In order to treat linear-quadratic problems with pointwise inequality constraints on the control, the multigrid approach is modified to solve subproblems generated by a primal-dual active set strategy (PDAS). Numerical experiments demonstrate the high efficiency of this approach due to mesh-independent convergence of both the outer PDAS method and the inner multigrid solver. The PDAS-multigrid method is incorporated in the sequential quadratic programming (SQP) framework. Inexact Newton techniques further enhance the computational efficiency. Globalization is implemented with a line search based on the augmented Lagrangian merit function. Numerical experiments highlight the efficiency of the resulting SQP-multigrid approach. In all cases, locally superlinear convergence of the SQP method is observed. In combination with the mesh-independent convergence rate of the inner solver, a solution method with optimal efficiency is obtained

Numerical Formulations For Attainable Region Analysis

Student Number : 9611112G - PhD thesis - School of Chemical and Metallurgical Engineering - Faculty of Engineering and the Built EnvironmentAttainable Region analysis is a chemical process synthesis technique that enables a design engineer to find process unit configurations that can be used to identify all possible outputs, by considering only the given feed specifications and permitted fundamental processes. The mathematical complexity of the attainable regions theory has so far been a major drawback in the implementation of this powerful technique into standard process design tools. In the past five years researchers focused on developing systematic methods to automate the procedure of identifying the set of all possible outputs termed the Attainable Regions. This work contributes to the development of systematic numerical formulations for attainable region analysis. By considering combinations of fundamental processes of chemical reaction, bulk mixing and heat transfer, two numerical formulations are proposed as systematic techniques for automation of identifying optimal process units networks using the attainable region analysis. The first formulation named the recursive convex control policy (RCC) algorithm uses the necessary requirement for convexity to approximate optimal combinations of fundamental processes that outline the shape of the boundary of the attainable regions. The recursive convex control policy forms the major content of this work and several case studies including those of industrial significance are used to demonstrate the efficiency of this technique. The ease of application and fast computational run-time are shown by assembling the RCC into a user interfaced computer application contained in a compact disk accompanying this thesis. The RCC algorithm enables identifying solutions for higher dimensional and complex industrial case studies that were previously perceived impractical to solve. The second numerical formulation uses singular optimal control techniques to identify optimal combinations of fundamental processes. This formulation also serves as a guarantee that the attainable region analysis conforms to Pontryagin’s maximum principle. This was shown by the solutions obtained using the RCC algorithm being consistent with those obtained by singular optimal control techniques

Systems and control : 21th Benelux meeting, 2002, March 19-21, Veldhoven, The Netherlands

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