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1

One-dimensional reflected bsdes with two barriers under logarithmic growth and applications

El Asri Brahim, Oufdil Khalid, Ourkiya Nacer

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 2

251--282

EN

We deal with the problem of existence and uniqueness of a solution for one-dimensional reflected backward stochastic differential equations with two strictly separated barriers when the generator has logarithmic growth |y| |ln |y|| + |z|√(|ln |z||) in the state variables y and z. The terminal value ξ and the obstacle processes (Lt)0≤t≤T and (Ut)0≤t≤T are Lp-integrable for a suitable p > 2. The main idea is to use the concept of local solution to construct a global one. As applications, we broaden the class of functions for which mixed zero-sum stochastic differential games admit an optimal strategy and the related double-obstacle partial differential equation problem has a unique viscosity solution.

2

Shape optimization for stationary Navier-Stokes equations

Halanay A., Tiba D.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1359-1374

EN

This work discusses geometric optimization problems governed by stationary Navier-Stokes equations. Optimal domains are proved to exist under the assumption that the family of admissible domains is bounded and satisfies the Lipschitz condition with a uniform constant, and in the absence of the uniqueness property for the state system. Through the parametrization of the admissible shapes by continuous functions defined on a larger universal domain, the optimization parameter becomes a control, i.e. an element of that family of continuous functions. The approximating extension technique via the penalization of the Navier-Stokes equation enables the approximation of the associated shape optimization problem by an optimal control problem. Results on existence and uniqueness are proved for the approximating problem and a gradient-type algorithm is indicated.

3

On modeling and optimal design of beam - bridge structures

Pedersen P., Pedersen N. L.

Foundations of Civil and Environmental Engineering

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2006

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No. 7

273-291

EN

To obtain black and white solutions (material or non-material) penalizations are applied, and due to problems of low density we can see a clear tendency toward solutions which more or less are truss or frame structures. Often, the accuracy of the finite element models for the continuum is then at its limits. For multiple load cases the formulation with a combination of individual load cases is in reality just as simple as for single load cases, but the design solution naturally depends on the selected combination factors, and we can illustrate this by a 3 D bridge example. It is still possible to obtain solutions by simple optimality criterion iterations which to a large extend, are used in this study. At first, the purpose of the presented paper is to make a comparison between optimal designs found by known methods for topology optimization of continuum structures and optimal designs of structures modeled as trusses. For a statically determined truss each bar can be designed independently and therefore must be fully stressed in an optimal design. We want to focus on the basic knowledge which gives an optimality criterion for single load eases with only a single constraint. Truss and continuum examples are analyzed, optimized, and evaluated to get further insight into the influence from the basic modeling, being truss or continuum. Stiffness as well as strength are important aspects of an optimal design.