scholarly journals Optimal Boundary Control of Non-Isothermal Viscous Fluid Flow

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 133 ◽  
Author(s):  
Evgenii S. Baranovskii ◽  
Anastasia A. Domnich ◽  
Mikhail A. Artemov
Keyword(s):  
Control Parameters ◽  
Marginal Function ◽  
Weak Formulation ◽  
Existence Of Weak Solutions ◽  
Viscous Fluid Flow ◽  
Lower Semi ◽  
Locally Lipschitz ◽  
Creeping Flows ◽  
Flow Domain ◽  
The Mathematical Model

We study an optimal control problem for the mathematical model that describes steady non-isothermal creeping flows of an incompressible fluid through a locally Lipschitz bounded domain. The control parameters are the pressure and the temperature on the in-flow and out-flow parts of the boundary of the flow domain. We propose the weak formulation of the problem and prove the existence of weak solutions that minimize a given cost functional. It is also shown that the marginal function of this control system is lower semi-continuous.

Solvability of free boundary problems for steady groundwater flow

2015 ◽  
Vol 26 (6) ◽  
pp. 821-847 ◽  
Author(s):  
A. Yu. BELIAEV
Keyword(s):  
Variational Principle ◽  
Free Boundary ◽  
Weak Solutions ◽  
Input Data ◽  
Free Boundary Problems ◽  
Boundary Problems ◽  
Phreatic Surface ◽  
Existence Of Weak Solutions ◽  
Lower Semi ◽  
Flow Domain

In this paper the free boundary problem for groundwater phreatic surface is represented in the form of a variational principle. It is proved that the flow domain Ω that solves the problem is a minimizer of some functional Λ(Ω). Weak solutions are introduced as minimizers of the lower semi-continuous regularization of Λ(⋅). Within this approach the existence of weak solutions is proved for a wide class of input data.

scholarly journals Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1300
Author(s):  
Evgenii S. Baranovskii ◽  
Vyacheslav V. Provotorov ◽  
Mikhail A. Artemov ◽  
Alexey P. Zhabko
Keyword(s):  
Nonlinear Boundary ◽  
Galerkin Approximation ◽  
Large Data ◽  
Temperature Dependent ◽  
Weak Formulation ◽  
Temperature Dependent Viscosity ◽  
Pipeline Network ◽  
Flux Boundary Conditions ◽  
Creeping Flows ◽  
Dependent Viscosity

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains.

The Multidirectional Mean Value Theorem in Banach Spaces

1997 ◽  
Vol 40 (1) ◽  
pp. 88-102 ◽  
Author(s):  
M. L. Radulescu ◽  
F. H. Clarke
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Locally Lipschitz Functions ◽  
Bump Function ◽  
Lipschitz Continuous ◽  
Lower Semi ◽  
Locally Lipschitz

AbstractRecently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

scholarly journals Some hemivariational inequalities in the Euclidean space

2019 ◽  
Vol 9 (1) ◽  
pp. 958-977 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Hemivariational Inequalities ◽  
Existence Of Weak Solutions ◽  
Lipschitz Continuous ◽  
Variational Structure ◽  
Locally Lipschitz ◽  
Sign Changing Solutions ◽  
Continuous Functionals

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

Sliding Mode Control of the Human Vestibular System

2012 ◽  
Author(s):  
Rachael McCarty ◽  
S. Nima Mahmoodi ◽  
Keith Williams
Keyword(s):  
Mathematical Model ◽  
Sliding Mode Control ◽  
Vestibular System ◽  
Head Movement ◽  
Sliding Mode ◽  
Sliding Mode Controller ◽  
Control Parameters ◽  
Mode Control ◽  
The Mathematical Model ◽  
The Brain

An original sliding mode controller is designed, based on an existing mathematical model for response control of the human vestibular system. The human vestibular system is located in the inner ear and significantly contributes to the functions of detecting head motion, maintaining balance and posture, and realizing gaze stabilization. The vestibular system sends signals to the brain to tell it how the head and body are moving, and the brain reacts by changing eye position accordingly. The nonlinearities of the vestibular system are not completely understood. The biggest nonlinearity is the nystagmus, a bouncing of the eyes to compensate for quick head movement. Another nonlinearity is that the quick phase does not start until head movement reaches a certain frequency. Considering these nonlinearities as well as the uncertainties of the system, sliding mode control a good choice for controlling the system. Several mathematical models of the human vestibular system are considered for use in the control design. The best model of those considered is chosen based on the models’ consideration of nonlinearities and their levels of complexity. The mathematical model used in this paper is a nonlinear transfer function. The output is controlled with a robust sliding mode controller. Results demonstrate the need to increase control parameters as frequency of the sinusoidal input increases to minimize overshoot error. However, since the human head cannot tolerate an infinitely large frequency input, control parameters also will necessarily be limited. Therefore, results show that the designed sliding mode robust controller is an effective mechanism for controlling the mathematical model of the human vestibular system.

A Study of the Effects of Baffles on Rotating Compressible Flows

1989 ◽  
Vol 56 (3) ◽  
pp. 710-712
Author(s):  
Max D. Gunzburger ◽  
Houston G. Wood ◽  
Rosser L. Wayland
Keyword(s):  
Fluid Dynamics ◽  
Mathematical Model ◽  
Compressible Flows ◽  
Numerical Examples ◽  
Thermal Boundary Conditions ◽  
Gas Centrifuge ◽  
Mass Injection ◽  
Flow Domain ◽  
The Mathematical Model ◽  
Element Algorithm

Onsager’s pancake equation for the fluid dynamics of a gas centrifuge is modified for the case of centrifuges with baffles which render the flow domain doubly connected. A finite element algorithm is used for solving the mathematical model and to compute numerical examples for flow fields induced by thermal boundary conditions and by mass injection and extraction.

Nonlinear Vibration Control of a Multi-Degree-of-Freedom Structure

2013 ◽  
Vol 311 ◽  
pp. 105-110
Author(s):  
Shueei Muh Lin
Keyword(s):  
Nonlinear Vibration ◽  
Passive Control ◽  
Structural Vibration ◽  
Control Parameters ◽  
Earthquake Excitation ◽  
Vibration Model ◽  
Highly Nonlinear ◽  
Conventional Structure ◽  
Nonlinear Vibration Control ◽  
The Mathematical Model

In this study, the nonlinear vibration model of structure with cross support is established. The conventional structure without cross support is linear and easy to be investigated. Unfortunately, its dynamic stability and vibration due to earthquake excitation are usually not acceptable. For suppressing the structural vibration the cross support composed of the elastic connecting bar and damper is considered here. This is a passive control design. Beside, due to the supporting arrangement, the mathematical model of the structure is highly nonlinear. In this study, the analytical solution for this system is derived. Further, the effects of control parameters on the vibration response are investigated.

ON THE EXISTENCE OF WEAK SOLUTIONS TO THE STATIONARY SEMICONDUCTOR EQUATIONS: THE CURRENT DRIVEN CASE

1999 ◽  
Vol 09 (01) ◽  
pp. 111-126
Author(s):  
J. NAUMANN
Keyword(s):  
Weak Solution ◽  
Semiconductor Device ◽  
Mixed Boundary ◽  
Diffusion Equations ◽  
Total Current ◽  
Mixed Boundary Value Problem ◽  
Drift Diffusion ◽  
Weak Formulation ◽  
Existence Of Weak Solutions ◽  
Semiconductor Equations

This paper is concerned with the stationary drift-diffusion equations of a semiconductor device where along a part of the device boundary the total current flux is prescribed. We introduce the weak formulation of this mixed boundary value problem and prove the existence of a weak solution.

scholarly journals 2-SYMMETRIC CRITICAL POINT THEOREMS FOR NON-DIFFERENTIABLE FUNCTIONS

2008 ◽  
Vol 50 (3) ◽  
pp. 447-466 ◽  
Author(s):  
PASQUALE CANDITO ◽  
ROBERTO LIVREA ◽  
DUMITRU MOTREANU
Keyword(s):  
Critical Point ◽  
Critical Values ◽  
Lipschitz Continuous ◽  
Unbounded Sequence ◽  
Differentiable Functions ◽  
Lower Semi ◽  
Locally Lipschitz ◽  
Critical Point Theorems ◽  
Locally Lipschitz Continuous

AbstractIn this paper, some min–max theorems for even andC1functionals established by Ghoussoub are extended to the case of functionals that are the sum of a locally Lipschitz continuous, even term and a convex, proper, lower semi-continuous, even function. A class of non-smooth functionals admitting an unbounded sequence of critical values is also pointed out.

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