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We analyze an optimal control problem governed by a rate-independent system in an abstract infinite-dimensional setting. The rate-independent system is characterized by a nonconvex stored energy functional, which depends on time via a time-dependent external loading, and by a convex dissipation potential, which is assumed to be bounded and positively homogeneous of degree one. The optimal control problem uses the external load as control variable and is constrained to normalized parametrized balanced viscosity solutions (BV solutions) of the rate-independent system. Solutions of this type appear as vanishing viscosity limits of viscously regularized versions of the original rate-independent system. Since BV solutions in general are not unique, as a main ingredient for the existence of optimal solutions we prove the compactness of solution sets for BV solutions. [ABSTRACT FROM AUTHOR]

We study a rate-independent system with non-convex energy in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized -solutions is obtained via vanishing viscosity in a suitable parameterized setting. In addition, we prove that the solution set is compact. [ABSTRACT FROM AUTHOR]

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example. [ABSTRACT FROM AUTHOR]

We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem. [ABSTRACT FROM AUTHOR]

This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Γ-convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance. [ABSTRACT FROM AUTHOR]

Abstract: We consider rate-independent evolutionary systems over a physical domain that are governed by simple hysteresis operators at each material point. For multiscale systems where denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit we are led to systems where the hysteresis operators at each macroscopic point is a generalized Prandtl–Ishlinskii operator. [Copyright &y& Elsevier]

This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff-Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Γ-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies. [ABSTRACT FROM AUTHOR]

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 Γ-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. [ABSTRACT FROM AUTHOR]

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 G-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. [ABSTRACT FROM AUTHOR]

The letter presents a penalized version of the time-discretization local minimization scheme first proposed by Efendiev and Mielke in 2006 to resolve time discontinuities in rate-independent systems with nonconvex energies. In order to penalize inequality constrains enforcing the local minimality, the Moreau–Yosida approximation is employed. We prove the convergence of time-discrete solutions to functions that are parametrized BV solutions of the time-continuous problem (in an abstract infinite-dimensional setting), provided that the discretization and approximation parameters are chosen appropriately. We test our scheme on a one-dimensional example and find a notable improvement compared with the original version. [ABSTRACT FROM AUTHOR]