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1

Certain convergence results for homogeneous singular young measures

Puchała Piotr

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 4

44--52

EN

We consider purely singular homogeneous Young measures associated with elements of sequences of piecewise constant functions and with limits of such sequences. We first consider a case when the limit of a such sequence is piecewise constant. The next point involves the sequences of bounded oscillating functions, divergent in the strong topology in L ∞ , but weakly∗ convergent to a homogeneous Young measure. We also present an example of a fast oscillating sequence, illustrating the result. In the presented results, generalizing to some extent known examples, we try to avoid advanced methods of functional analysis that are usually used when solving problems of this type.

2

On a certain embedding in the space of measures

Puchała Piotr

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 2

53--63

EN

We take under consideration Young measures - objects that can be interpreted as generalized solutions of a class of certain nonconvex optimization problems arising among others in nonlinear elasticity or micromagnetics. They can be looked at from several points of view. We look at Young measures as at a class of weak* measurable, measure-valued mappings and consider the basic existence theorem for them. On the basis of this theorem, an imbedding of the set of bounded Borel functions into the set of Young measures is defined. Using the weak* denseness of the set of Young measures associated with simple functions in the set of Young measures, it is shown that this imbedding assigns the Young measure associated with any bounded Borel function.

3

A few remarks on an embedding into the set of measures

Puchała Piotr

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 4

77--85

EN

We continue considerations concerning Young measures associated with bounded measurable functions from a recent article. We look at them as at the weak* measurable, measure-valued mappings. We show examples explaining that we cannot regard a Young measure (i.e. a weak* -measurable mapping) δu(x) as an explicit form of a Young measure associated with a function u. We also consider convergence of the sequences of Young measures.

4

Remarks about discrete Young measures and their Monte Carlo simulation

Grzybowski A., Puchała P.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 2

13--20

EN

This article is devoted to the problem of simulation of random variables distributed according to Young measures associated with piecewise affine functions determined on bounded intervals. We start with simple functions which can take on a finite number of different values with inverse images being the intervals or their unions. We present some formal results connected with related discrete Young measures and propose an algorithm for generating random variables having such distributions. Next, based on these results we introduce an algorithm designed for approximation of Young measures in various, more general situations. We also present an example where a Young measure associated with a piecewise affine function is approximated with the help of computer simulations. In this benchmarking problem the theoretical results are compared with the ones obtained in the Monte Carlo experiment.

5

Bang-bang controls in the singular perturbations limit

Artstein Z.

Control and Cybernetics

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2005

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Vol. 34, no 3

645-663

EN

A general form of the dynamics obtained as a limit of trajectories of singularly perturbed linear control systems is presented. The limit trajectories are described in terms of probability measure-valued maps. This allows to determine the extent to which the bang-bang principle for linear control systems is carried over to the singular limit.

6

Miary Younga i ich zastosowania w mikromechanice i optymalizacji. Część I. Podstawy matematyczne

Bielski W., Kruglenko E., Telega J. J.

Matematyka Stosowana : matematyka dla społeczeństwa

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2003

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Nr 4

90-138

7

Compactness criteria for the stable topology

Raynaud de Fitte P.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 4

343-363

EN

Adapting a general result of Topsoe [32], we prove a compactness criterion for the stable topology on the set of measures on the product of a measurable space and a Suslin (nonnecessarily regular) topological space. We also extend a compactness criterion of Jacod and Memin [17] and we apply these results to the case of Young measures.

8

Three boundary value problems for second order differential inclusions in Banach spaces

Azzam D., Castaing C., Thibault L.

Control and Cybernetics

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2002

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Vol. 31, no 3

659-693

EN

The paper studies, in the context of Banach spaces, the problem of three boundary conditions for both second order differential inclusions and second order ordinary differential equations. The results are obtained in several new settings of Sobolev-type spaces involving Bochner and Pettis integrals. Some classes of second order multivalued evolution equations associated with m-accretive operators are also considered. Applications to some control problems are provided with the help of narrow convergence for Young measures.

9

Existence in optimal control problems of certain Fredholm integral equations

Roubicek T., Schmidt W.

Control and Cybernetics

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2001

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Vol. 30, no 3

303-322

EN

Existence of an optimal control for certain systems governed by nonlinear Fredholm integral equations that are of a Hammerstein type wlith respect to the control is proved under convexity of the orientor field by using a relaxed problem. Then, this convexity assumption is put away by finer analysis of the maximum principle. Illustrative examples are presented.