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End date: 2022 × Start date: 2020 × End date: 2019 × Start date: 2010 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: optimal control ×

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Now showing 1 - 4 of 4
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    Stability of the solution set of quasi-variational inequalities and optimal control
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.
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    Optimal control of geometric partial differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hintermüller, Michael; Keil, Tobias
    Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of nondegenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint.
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    Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Gräßle, Carmen; Hintermüller, Michael; Hinze, Michael; Keil, Tobias
    We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.
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    A distributed control problem for a fractional tumor growth model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24.) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl. 28 (2019), 343--375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.