scholarly journals Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 483
Author(s):  
Marina Plekhanova ◽  
Guzel Baybulatova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Evolution Equations ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Equations ◽  
Initial Boundary Value ◽  
Degenerate Evolution Equations

A theorem on unique solvability in the sense of the strong solutions is proved for a class of degenerate multi-term fractional equations in Banach spaces. It applies to the deriving of the conditions on unique solution existence for an optimal control problem to the corresponding equation. Obtained results are used to an optimal control problem study for a model system which is described by an initial-boundary value problem for a partial differential equation.

scholarly journals Regularized optimal control problem for a beam vibrating against an elastic foundation

2015 ◽  
Vol 63 (1) ◽  
pp. 53-71
Author(s):  
Igor Bock ◽  
Mária Kečkemétyová
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elastic Foundation ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value

Abstract We deal with an optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing the perpendicular vibrations of a clamped beam against a u elastic foundation. A variable thickness of a beam plays the role of a control variable. The original equation for the deflection is regularized in order to derive necessary optimality conditions

Optimal control problem for the equation of vibrations of an elastic plate

2018 ◽  
Vol 25 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Hamlet F. Guliyev ◽  
Khayala I. Seyfullaeva
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elastic Plate ◽  
Control Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Necessary Condition ◽  
Necessary Condition Of Optimality ◽  
Initial Boundary Value

AbstractAn optimal control problem for the vibration equation of an elastic plate is considered when the control function is included in the coefficient of the highest order derivative and the right-hand side of the equation. The solvability of the initial boundary value problem is shown, the theorem on the existence of an optimal control is proved and a necessary condition of optimality in the form of an integral equation is obtained.

scholarly journals An Optimal Control Problem for A Viscoelastic Plate in a Dynamic Contact with an Obstacle

2018 ◽  
Vol 71 (1) ◽  
pp. 27-37
Author(s):  
Igor Bock ◽  
Mária Kečkemétyová
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Variable ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dynamic Contact ◽  
Viscoelastic Plate ◽  
Initial Boundary Value ◽  
Thickness Function

Abstract We deal with an optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing the perpendicular vibrations of a simply supported anisotropic viscoelastic plate against a rigid obstacle. A variable thickness of a plate plays the role of a control variable. We verify the existence of an optimal thickness function.

Optimal control of thermal processes with phase transitions

2021 ◽  
Author(s):  
Yury Evtushenko ◽  
Vladimir Zubov ◽  
Anna Albu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Crystallization Process ◽  
Automatic Differentiation ◽  
Phase Interface ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Phase ◽  
Metal Solidification

The optimal control of the metal solidification process in casting is considered. Quality of the obtained detail greatly depends on how the crystallization process proceeds. It is known that to obtain a model of a good quality it is desirable that the phase interface would be as close as possible to a plane and that the speed of its motion would be close to prescribed. The proposed mathematical model of the crystallization process is based on a three dimensional two phase initial-boundary value problem of the Stefan type. The velocity of the mold in the furnace is used as the control. The control satisfying the technological requirements is determined by solving the posed optimal control problem. The optimal control problem was solved numerically using gradient optimization methods. The effective method is proposed for calculation of the cost functional gradient. It is based on the fast automatic differentiation technique and produces the exact gradient for the chosen approximation of the optimal control problem.

Mixed Control Problem for the Linearized Quasi-Stationary Phase Field System of Equations

2016 ◽  
Vol 845 ◽  
pp. 170-173 ◽  
Author(s):  
Marina Plekhanova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Phase Field ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Operator Differential Equation ◽  
Field Equations ◽  
Mixed Control ◽  
Phase Field Equations

Conditions are obtained for unique solution existence of a mixed control problem without taking in account control expenses for a system that described by an initial-boundary value problem for the linearized quasi-stationary system of phase field equations. The problem is reduced to an optimal control problem for operator differential equation of first order in abstract space with degenerate operator at derivative using start and distributed controls simultaneously. The theorem on the unique solvability of this problem is applied to studying of optimal control problem for the phase field equations system.

Inverse extremum problem for a model of endovenous laser ablation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrey Kovtanyuk ◽  
Alexander Chebotarev ◽  
Alena Astrakhantseva
Keyword(s):  
Optimal Control ◽  
Laser Ablation ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Extremum Problem ◽  
Transfer Model ◽  
Conductive Heat ◽  
Endovenous Laser Ablation

AbstractAn inverse extremum problem (optimal control problem) for a quasi-linear radiative-conductive heat transfer model of endovenous laser ablation is considered. The problem is to find the powers of the source spending on heating the fiber tip and on radiation. As a result, it provides a given temperature distribution in some part of the model domain. The unique solvability of the initial-boundary value problem is proved, on the basis of which the solvability of the optimal control problem is shown. An iterative algorithm for solving the optimal control problem is proposed. Its efficiency is illustrated by a numerical example.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

scholarly journals A Class of Fractional Degenerate Evolution Equations with Delay

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1700
Author(s):  
Amar Debbouche ◽  
Vladimir E. Fedorov
Keyword(s):  
Unique Solvability ◽  
Evolution Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Systems Of Equations ◽  
Fractional Differential ◽  
Initial Boundary Value ◽  
Degenerate Type ◽  
Degenerate Evolution Equations ◽  
Equations With Delay

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappings. The obtained theory via its abstract results is applied to the research of initial-boundary value problems for both Scott–Blair and modified Sobolev systems of equations with delays.

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