FG Optimale Steuerung
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We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal counterexample to various statements in convex analysis in which completeness is required.
We investigate a broad class of integer optimal control problems with vector-valued controls and switching regularization using a total variation functional involving the p-norm, which influences the structure of a solution. We derive optimality conditions of first and second order for the integer optimal control problem via a switching-point reformulation. For the numerical solution, we use a trust-region method utilizing Bellman’s optimality principle for the subproblems. We will show convergence properties of the method and highlight the algorithm’s efficacy on some benchmark examples.
We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalized problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving, for example, the positive part function max(0, ⋅). Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem.
A solution algorithm for a special class of optimal control problems subject to an ordinary differential equation is proposed. The controls possess a continuous-or-off structure and are priced by a convex function. Additionally a total variation regularization is applied to penalize switches. Our solution method combines a trust-region method and a proximal gradient method. The subproblems are solved via Bellman's optimality principle. Convergence with respect to a criticality measure is proven. As a numerical example, we solve a simple optimal control problem involving an SIR model.
An optimal control problem in the space of Borel measures governed by the Poisson equation is investigated. The characteristic feature of the problem under consideration is the Tikhonov regularization term in form of the transportation distance of the control to a given prior. Existence of optimal solutions is shown and first-order necessary optimality conditions are derived. The latter are used to deduce structural a priori information about the optimal control and its support based on properties of the associated optimal transport plan.
We address optimal control problems on the space of measures for an objective containing a smooth functional and an optimal transport regularization. That is, the quadratic Monge-Kantorovich distance between a given prior measure and the control is penalized in the objective. We consider optimality conditions and reparametrize the problem using the celebrated structure theorem by Brenier. The optimality conditions can be formulated as a piecewise differentiable equation. This is utilized to formulate solution algorithms and to analyze their local convergence properties. We present a numerical example to illustrate the theoretical findings.
Proximal gradient methods are a popular tool for the solution of structured, nonsmooth minimization problems. In this work, we investigate an extension of the former to general Banach spaces and provide worst-case convergence rates for, both, convex and nonconvex, problem instances. Moreover, assuming additional regularity properties of stationary points, linear rates of convergence are derived. The theoretical results are illustrated for bang-bang type optimal control problems with partial differential equations which we study in the space of Radon measures. An efficient implementation of the resulting -proximal gradient method is given and its performance is compared to standard -proximal gradient as well as Frank-Wolfe methods. The paper is complemented by discussing the relationship among different regularity properties as well as by providing a novel characterization of the Polyak--Łojasiewicz--Kurdyka property via second-order conditions involving weak* second subderivatives.