On modeling thin inclusions in elastic bodies with a damage parameter

2018 ◽  
Vol 24 (9) ◽  
pp. 2742-2753 ◽  
Author(s):  
A. M. Khludnev
Keyword(s):  
Optimal Control ◽  
Nonlinear Boundary ◽  
Equilibrium Problems ◽  
Control Function ◽  
Nonlinear Boundary Conditions ◽  
Energy Functional ◽  
Damage Parameter ◽  
Elastic Bodies ◽  
The Matrix ◽  
Crack Faces

In this paper, we analyze equilibrium problems for 2D elastic bodies with two thin inclusions in the presence of damage. A delamination of the inclusions from the matrix is assumed, thus forming a crack between the elastic body and the inclusions. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. Weak and strong formulations of the problems are discussed. The paper provides an asymptotic analysis with respect to the damage parameter. An optimal control problem is analyzed with a cost functional to be equal to the derivative of the energy functional with respect to the crack length, and the damage parameter being a control function.

Optimal control of linear systems with fractional derivatives

2018 ◽  
Vol 21 (1) ◽  
pp. 134-150 ◽  
Author(s):  
Ivan Matychyn ◽  
Viktoriia Onyshchenko
Keyword(s):  
Optimal Control ◽  
Linear Systems ◽  
Control Function ◽  
Support Functions ◽  
Jordan Canonical Form ◽  
Terminal Set ◽  
Control Functions ◽  
The Matrix ◽  
Time Optimal ◽  
Control Of Linear Systems

Abstract Problem of time-optimal control of linear systems with fractional Caputo derivatives is examined using technique of attainability sets and their support functions. A method to construct a control function that brings trajectory of the system to a strictly convex terminal set in the shortest time is elaborated. The proposed method uses technique of set-valued maps and represents a fractional version of Pontryagin’s maximum principle. A special emphasis is placed upon the problem of computing of the matrix Mittag-Leffler function, which plays a key role in the proposed methods. A technique for computing matrix Mittag-Leffler function using Jordan canonical form is discussed, which is implemented in the form of a MATLAB routine. Theoretical results are supported by examples, in which the optimal control functions, in particular of the “bang-bang” type, are obtained.

scholarly journals Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff–Love plates with a crack

2019 ◽  
Vol 24 (12) ◽  
pp. 3743-3752 ◽  
Author(s):  
Nyurgun Lazarev ◽  
Hiromichi Itou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Rigid Inclusion ◽  
Equilibrium Problems ◽  
Location Parameter ◽  
Continuous Functional ◽  
Existence Of A Solution ◽  
Crack Faces ◽  
Variational Statement

A non-linear model describing the equilibrium of a cracked plate with a volume rigid inclusion is studied. We consider a variational statement for the Kirchhoff–Love plate satisfying the Signorini-type non-penetration condition on the crack faces. For a family of problems, we study the dependence of their solutions on the location of the inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on a suitable Sobolev space. For this problem, the location parameter of the inclusion serves as a control parameter. We prove continuous dependence of the solutions with respect to the location parameter and the existence of a solution of the optimal control problem.

Dynamics Response Control of a Mindlin-Type Beam

2017 ◽  
Vol 17 (03) ◽  
pp. 1750039 ◽  
Author(s):  
Kenan Yildirim ◽  
Seda G. Korpeoglu ◽  
Ismail Kucuk
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Maximum Principle ◽  
Control Problem ◽  
Control Function ◽  
Initial Boundary ◽  
Response Control ◽  
Adjoint Variables ◽  
Dynamics Response ◽  
Optimal Control Function

Optimal boundary control for damping the vibrations in a Mindlin-type beam is considered. Wellposedness and controllability of the system are investigated. A maximum principle is introduced and optimal control function is obtained by means of maximum principle. Also, by using maximum principle, control problem is reduced to solving a system of partial differential equations including state, adjoint variables, which are subject to initial, boundary and terminal conditions. The solution of the system is obtained by using MATLAB. Numerical results are presented in table and graphical forms.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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