Filip Rindler

Optimal Control for Nonconvex Rate-Independent Evolution Processes

Energetic solutions to rate-independent systems allow for an effective modeling of many physical systems displaying hysteretic effects, e.g., phase transformations in shape-memory alloys, elastoplasticity, and ferroelectricity. For some engineering applications, optimal control of such systems is desirable. We establish existence results for these continuous-time optimal control problems by a combination of the direct method with

Approximation of Rate-Independent Optimal Control Problems

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Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

-convergence arguments. Applicability to the common situation of a controlled external loading is demonstrated and a concrete partial differential inclusion as well as an academic model of the foreign exchange market including trading costs are investigated.

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

This work introduces an abstract approximation scheme for optimal control problems in the energetic theory of rate-independent systems. This scheme builds upon the usual temporal semidiscretization and a

Characterization of Generalized Young Measures Generated by Symmetric Gradients

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Differential Inclusions and Young Measures Involving Prescribed Jacobians

-convergence approximation of the functionals (in a full discretization, this approximation consists of a sequence of discrete functionals defined on finer and finer discrete spaces). The main new result of this work is a rigorous convergence analysis for this scheme. Interestingly, it turns out that the usual time discretization is incomplete as a basis for a fully discrete approximation of the optimal control problem, and a new nonlocal condition has to be introduced. Since this new condition is necessary and sufficient for a one-to-one correspondence of discrete-time and continuous-time solutions to rate-independent systems, we also gain a new theoretical insight, which might be of independent interest. At the heart of the derivation are so-called reverse approximation results, for which we provide extensions and refinements. To give a concrete example and to demonstrate how the abstract theory can be applied in a concrete problem, we present a full discretization with finite elements of the optimal control problem for a rate-independent partial differential inclusion.

Regularity and approximation of strong solutions to rate-independent systems

We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form

Generalized Young Measures

\begin{array}{ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). \end{array}