Optimal Control for a Nonlocal Model of Non-Newtonian Fluid Flows

This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. The control parameter is the external force. We discuss both strong and weak solutions. Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. Moreover, a necessary condition for the existence of strong solutions is derived. Simultaneously, we introduce the concept of the marginal function and study its properties. In particular, it is shown that the marginal function of this control system is lower semicontinuous with respect to the directed Hausdorff distance.

Download Full-text

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

Numerical Linear Algebra with Applications ◽

10.1002/nla.2404 ◽

2021 ◽

Author(s):

Britta Schmitt ◽

Boris N. Khoromskij ◽

Venera Khoromskaia ◽

Volker Schulz

Keyword(s):

Optimal Control ◽

Elliptic Operator ◽

Optimal Control Problems ◽

Three Dimensional ◽

Variable Coefficients ◽

Control Problems ◽

Tensor Method

Download Full-text

A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000010432 ◽

1995 ◽

Vol 36 (3) ◽

pp. 261-273 ◽

Cited By ~ 3

Author(s):

Mohammad A. Kazemi

Keyword(s):

Optimal Control ◽

Partial Differential Equations ◽

Differential Equations ◽

Optimal Control Problems ◽

Fréchet Derivative ◽

Necessary Condition ◽

Control Problems ◽

Partial Differential ◽

Frechet Derivative ◽

First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

Download Full-text

Optimal control of three-dimensional steamflooding processes

International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts ◽

10.1016/0148-9062(95)90146-9 ◽

1995 ◽

Vol 32 (1) ◽

pp. A20

Keyword(s):

Optimal Control ◽

Three Dimensional

Download Full-text

Trapping of waves by a submerged elliptical torus

Journal of Fluid Mechanics ◽

10.1017/s0022112001007327 ◽

2002 ◽

Vol 456 ◽

pp. 277-293 ◽

Cited By ~ 13

Author(s):

M. McIVER ◽

R. PORTER

Keyword(s):

Fluid Velocity ◽

Three Dimensional ◽

Mode Frequency ◽

The Body ◽

Necessary Condition ◽

Numerical Evidence ◽

Side Condition ◽

Trapped Mode ◽

Specific Geometry ◽

The Time Domain

An investigation is made into the trapping of surface gravity waves by totally submerged three-dimensional obstacles and strong numerical evidence of the existence of trapped modes is presented. The specific geometry considered is a submerged elliptical torus. The depth of submergence of the torus and the aspect ratio of its cross-section are held fixed and a search for a trapped mode is made in the parameter space formed by varying the radius of the torus and the frequency. A plane wave approximation to the location of the mode in this space is derived and an integral equation and a side condition for the exact trapped mode are obtained. Each of these conditions is satisfied on a different line in the plane and the point at which the lines cross corresponds to a trapped mode. Although it is not possible to locate this point exactly, because of numerical error, existence of the mode may be inferred with confidence as small changes in the numerical results do not alter the fact that the lines cross.If the torus makes small vertical oscillations, it is customary to try to express the fluid velocity as the gradient of the so-called heave potential, which is assumed to have the same time dependence as the body oscillations. A necessary condition for the existence of this potential at the trapped mode frequency is derived and numerical evidence is cited which shows that this condition is not satisfied for an elliptical torus. Calculations of the heave potential for such a torus are made over a range of frequencies, and it is shown that the force coefficients behave in a singular fashion in the vicinity of the trapped mode frequency. An analysis of the time domain problem for a torus which is forced to make small vertical oscillations at the trapped mode frequency shows that the potential contains a term which represents a growing oscillation.

Download Full-text

Near Optimality of Linear Delayed Doubly Stochastic Control Problem

Mathematical Problems in Engineering ◽

10.1155/2021/4487092 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Jie Xu ◽

Ruiqiang Lin

Keyword(s):

Optimal Control ◽

Time Delay ◽

Control Problem ◽

Delay System ◽

Necessary Condition ◽

Linear Quadratic ◽

The Maximum Principle ◽

Doubly Stochastic ◽

Convex Control ◽

Near Optimality

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.

Download Full-text

The Yang-Mills heat equation with finite action in three dimensions

Memoirs of the American Mathematical Society ◽

10.1090/memo/1349 ◽

2022 ◽

Vol 275 (1349) ◽

Author(s):

Leonard Gross

Keyword(s):

Heat Equation ◽

Three Dimensional ◽

Strong Solutions ◽

Three Dimensions ◽

Dimensional Manifold ◽

Uniqueness Of Solutions ◽

Reconstruction Procedure ◽

Content Type ◽

Yang Mills ◽

Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

Download Full-text