scholarly journals On a Limiting Passage as the Thickness of a Rigid Inclusions in an Equilibrium Problem for a Kirchhoff-Love Plate with a Crack

2021 ◽  
pp. 28-41
Author(s):  
Nyurgun P Lazarev ◽  
Galina M Semenova ◽  
Natalya A. Romanova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Rigid Inclusion ◽  
Three Dimensional ◽  
Median Plane ◽  
Variational Problems ◽  
Initial State ◽  
Fixed Part ◽  
Thickness Parameter

The paper considers equilibrium models of Kirchhoff-Love plates with rigid inclusions of two types. The first type of inclusion is described by three-dimensional sets, the second one corresponds to a cylindrical rigid inclusion, which is perpendicular to the plate’s median plane in the initial state. For both models, we suppose that there is a through crack along a fixed part of the inclusion’s boundary. On the crack non-penetration conditions are prescribed which correspond to a certain known configuration bending near the crack. The uniqueness solvability of a new problems for a Kirchhoff-Love plate with a flat rigid inclusion is proved. It is proved that when a thickness parameter tends to zero, the problem for a flat rigid inclusion can be represented as a limiting task for a family of variational problems concerning the inclusions of the first type. A solvability of an optimal control problem with a control given by the size of inclusions is proved

Symmetric Control Systems

2021 ◽  
pp. 37-51
Author(s):  
A.I. Diveev ◽  
E.A. Sofronova
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Systems ◽  
Control Object ◽  
Terminal State ◽  
Initial State ◽  
Phase Constraints ◽  
The Mathematical Model

The paper focuses on the properties of symmetric control systems, whose distinctive feature is that the solution of the optimal control problem for an object, the mathematical model of which belongs to the class of symmetric control systems, leads to the solution of two problems. The first optimal control problem is the initial one; the result of its solution is a function that ensures the optimal movement of the object from the initial state to the terminal one. In the second problem, the terminal state is the initial state, and the initial state is the terminal state. The complexity of the problem being solved is due to the increase in dimension when the models of all objects of the group are included in the mathematical model of the object, as well as the emerging dynamic phase constraints. The presence of phase constraints in some cases leads to the target functional having several local extrema. A theorem is proved that under certain conditions the functional is not unimodal when controlling a group of objects belonging to the class of symmetric systems. A numerical example of solving the optimal control problem with phase constraints by the Adam gradient method and the evolutionary particle swarm method is given. In the example, a group of two symmetrical objects is used as a control object

scholarly journals GENERATION OF AN OPTIMAL LOW-ALTITUDE TRAJECTORY FOR A FIXED-WING UNMANNED AERIAL VEHICLE IN A MOUNTAINOUS AREA

Aviation ◽  
2021 ◽  
Vol 25 (2) ◽  
pp. 115-122
Author(s):  
Hossein Maghsoudi ◽  
Amirreza Kosari Kosari
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Planning ◽  
Optimal Trajectory ◽  
Three Dimensional ◽  
Direct Collocation ◽  
Aerial Vehicle ◽  
Low Altitude ◽  
Optimal Trajectory Planning

In this study, the three-dimensional optimal trajectory planning of an unmanned fixed-wing aerial vehicle was investigated for Terrain Following – Terrain Avoidance (TF-TA) purposes using the Direct Collocation method. For this purpose, firstly, the appropriate equations representing the translational movement of the aircraft were described. The three-dimensional optimal trajectory planning of the flying vehicle was formulated in the TF-TA manoeuvre as an optimal control problem. The terrain profile, as the main allowable height constraint was modelled using the Fractal Generation Method. The resulting optimal control problem was discretized by applying the Direct Collocation numerical technique and then, was transformed into a Nonlinear Programming Problem (NLP). The efficacy of the proposed method was demonstrated by extensive simulations, and it was particularly verified that the purposed approach can produce a solution satisfying almost all the performance and environmental constraints encountering in a low -altitude flight.

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.

scholarly journals Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

2020 ◽  
Vol 6 (4) ◽  
pp. 743-768
Author(s):  
Yuri Alexandrovich Kostikov ◽  
Alexander Mikhailovich Romanenkov
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Heat Equation ◽  
Control Problem ◽  
Three Dimensional ◽  
Type Equation ◽  
Parabolic Type ◽  
Multidimensional Case ◽  
Differential Problem ◽  
Difference Problem

This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

REDUCED MODEL OF PLASMA DISCHARGE CONTROL IN TOKAMAK WITH IRON CORE

2020 ◽  
Vol 7 (3) ◽  
pp. 11-22
Author(s):  
VALERY ANDREEV ◽  
◽  
ALEXANDER POPOV
Keyword(s):  
Optimal Control ◽  
Inverse Problem ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Nonlinear Behavior ◽  
Plasma Discharge ◽  
Reduced Model ◽  
Iron Core ◽  
Power Sources ◽  
Real Power

A reduced model has been developed to describe the time evolution of a discharge in an iron core tokamak, taking into account the nonlinear behavior of the ferromagnetic during the discharge. The calculation of the discharge scenario and program regime in the tokamak is formulated as an inverse problem - the optimal control problem. The methods for solving the problem are compared and the analysis of the correctness and stability of the control problem is carried out. A model of “quasi-optimal” control is proposed, which allows one to take into account real power sources. The discharge scenarios are calculated for the T-15 tokamak with an iron core.

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