Adjoint Boundary Value Problem Research Articles

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Quaternary Continuous Classical Boundary Optimal Control Problem Dominating by Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem dominated by a quaternary linear parabolic boundary value problem. Under suitable assumptions and with a given quaternary continuous classical boundary control vector, the existence theorem for a unique quaternary state vector solution to the weak form is stated and demonstrated via the method of Galerkin. Furthermore, the continuity of the Lipschitz operator between the state vector solution to the weak form for the dominating equations and the corresponding are proved. The existence of a quaternary continuous classical boundary optimal control vector is stated and demonstrated under suitable Assumptions. The mathematical formulation for the quaternary adjoint boundary value problem associated with each considered boundary value problem is obtained and the Frѐchet derivative for the objective function is derived. Finally, the necessary conditions for the optimality theorem of the problem are stated and demonstrated.

Solvability for Optimal Classical Continuous Control Problem Controlling by Quaternary Hyperbolic Boundary Value Problem

This work is concerned with studying the solvability for optimal classical continuous control quaternary vector problem that controls by quaternary linear hyperbolic boundary value problem. The existence of the unique quaternary state vector solution for the quaternary linear hyperbolic boundary value problem is studied and demonstrated by employing the method of Galerkin, where the classical continuous control quaternary vector is Known. Also, the existence theorem of an optimal classical continuous control quaternary vector related to the quaternary linear hyperbolic boundary value problem is demonstrated. The existence of a unique solution to the adjoint quaternary linear hyperbolic boundary value problem associated with the quaternary linear hyperbolic boundary value problem is formulated and studied. The directional derivative for the cost functional is derived. Finally, the necessary optimality theorem for the optimal classical continuous control quaternary vector is proved.

Towards photophoresis with the generalized Lorenz-Mie theory

Based on the adjoint boundary value problem proposed decades ago by Zulehner and Rohatschek [1], analytic and closed-form expressions for the photophoretic forces exerted by arbitrary-shaped beams on homogeneous and low-loss spherical particles is derived in both the free molecular and slip flow regimes. To do so, the asymmetry vector for arbitrary refractive index particles is explicitly calculated by expanding the internal electromagnetic fields with the aid of the generalized Lorenz-Mie theory (GLMT). The approach here proposed is, to the best of the authors’ knowledge, the first systematic attempt to incorporate the GLMT stricto sensu into the field of photophoresis and might as well be extended, e.g. to spheroids and find important applications, among others, in optical trapping and manipulation of microparticles, in geoengineering, particle levitation, optical trap displays and so on.

ОБРАТНАЯ ЗАДАЧА ТИПА УПРАВЛЕНИЯ ОБ ОПРЕДЕЛЕНИИ СТАРШЕГО КОЭФФИЦИЕНТА ОДНОМЕРНОГО ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ

In this paper, we consider one inverse control-type problem of determining the leading coefficient of a one-dimensional parabolic equation. The problem under consideration is a variational statement of a coefficient inverse problem for a parabolic equation. The sought for coefficient of the parabolic equation depends on the spatial variable. An integral boundary condition is set for the parabolic equation. The desired highest coefficient of the parabolic equation plays role of the control function, which is an element of the Sobolev space. The set of admissible control functions belong to the Sobolev space. The objective functional for the control problem is compiled based on the integral overdetermination condition set in the inverse problem. This condition may be interpreted as tasks of a weighted mean of the solution of the equation under consideration as per time variable. The solution of the boundary value problem for a parabolic equation, for each given control function, is defined as a generalized solution from the Sobolev space. The existence of a solution of the considered inverse control-type problem is proved. An adjoint boundary value problem for the control problem under study is introduced. The Frechet differentiability of the objective functional in a set of admissible control functions is proved. In addition, an auxiliary boundary value problem is introduced; and using the solution of this problem, an expression for the gradient of the objective functional is found. The necessary optimality condition for the admissible control function is obtained.

Kronecker Product of matrices and their applications to self-adjoint two-point boundary value problems associated with first order matrix differential systems

In this paper, we shall be concerned with Kronecker product or Tensor product of matrices and develop their properties in a systematic way. The properties of the Kronecker product of matrices is used as a tool to establish existence and uniqueness of solutions to two-point boundary value problems associated with system of first order differential systems. A new approach is described to solve the Kronecker product linear systems and establish best least square solutions to the problem. Several interesting examples are given to highlight the importance of Kronecker product of matrices. We present adjoint boundary value problems and deduce a set of necessary and sufficient conditions for the Kronecker product boundary value problem to be self-adjoint.

A Shape Optimisation with the Isogeometric Boundary Element Method and Adjoint Variable Method for the Three-Dimensional Helmholtz Equation

This paper presents a shape optimisation system to design the shape of an acoustically-hard object in the three-dimensional open space. The boundary element method (BEM) is suitable to analyse such an exterior field. However, the conventional BEM, which is based on piecewise polynomial shape and approximate (interpolation) functions, can require many design variables because they are usually chosen as a part of the nodes of the underlying boundary element mesh. In addition, it is not easy for the conventional method to compute the gradient of the sound pressure on the surface, which is necessary to compute the shape derivative of our interest, of a given object. To overcome these issues, we employ the isogeometric boundary element method (IGBEM), which was developed in our previous work. With using the IGBEM, we can design the shape of surfaces through control points of the NURBS surfaces of the target object. We integrate the IGBEM with the nonlinear programming software through the adjoint variable method (AVM), where the resulting adjoint boundary value problem can be also solved by the IGBEM with a slight modification. The numerical verification and demonstration validate our shape optimisation framework.

Features of search for complex solutions of non-self-adjoint boundary value problems of electrodynamics

The features of finding of complex solutions of the dispersion equations of guiding electrodynamic structures waves described by non-self-adjoint operators have been considered. These solutions are found using the combined numerical method for complex roots finding combining the Muller method and the phase variation method. A brief explanation of the combined search method essence has been given, and its advantages have been described. As examples, structures with simple geometry are selected in order to avoid difficult mathematical expressions and focus on the process of complex solutions finding. The guiding electrodynamic structures both without losses and with absorbing metal films have been considered in the approximation of the surface current method and based on its real thickness. A new approach to evaluate the physical correctness and accuracy of the results of solving dispersion problems set in an open form for guiding structures described by non-self-adjoint operators has been shown – by the tendency to zero of the power flow of proper complex waves on average over a period through the cross section of a non-dissipative guide structure. The formulation and the solution features of the adjoint boundary value problem for a round open dielectric waveguide have been considered. The existence possibility of the adjoint waves in this waveguide has been shown.

On the integral equation of an adjoint boundary value problem of heat conduction

On the integral equation of an adjoint boundary value problem of heat conduction

Deflection of axially functionally graded rectangular plates by Green\'s function method

This paper deals with the static analysis of axially functionally graded rectangular plates. It is assumed that the flexural rigidity of the plate varies exponentially along one of the plate's in-plane dimensions. Both an analytical approach and a numerical method are utilized to solve the problem. The analytical solution is obtained by using the Green's function method. To employ this approach, the adjoint boundary value problem is established. Then, exact solutions for deflection of the plate for different boundary conditions are found. In another way, a finite element formulation for the problem is developed. In order to demonstrate the validity of the Authors' formulation, the results obtained via both mentioned schemes are compared with each other for functionally graded plates and with results of previously published works for homogeneous plates. The effect of plate parameters on the response of the plate is also investigated. To remind the research background, a brief review on the application of Green's function method in plates' analysis and functionally graded plates is also presented.

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ℝ3 we consider the stationary Maxwell equations for a function u with values in ℝ3 subject to a nonhomogeneous condition (u, v)x = u0 on the boundary, where v is a given vector field and u0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.

Boundary Value Problem For a Sturm-Liouville Operator With Piecewise Continuous Coefficient

In this paper, a self adjoint boundary value problem with a piecewise continuous coefficient on the positive half line [0 , ∞ ) is considered. The resolvent operator is constructed and the expansion formula with respect to eigenfunctions or equivalently Parseval equality is obtained. The spectrum of the operator is discussed.

About Dirichlet boundary value problem for the heat equation in the infinite angular domain

Abstract In this paper it is established that in an infinite angular domain for Dirichlet problem of the heat conduction equation the unique (up to a constant factor) non-trivial solution exists, which does not belong to the class of summable functions with the found weight. It is shown that for the adjoint boundary value problem the unique (up to a constant factor) non-trivial solution exists, which belongs to the class of essentially bounded functions with the weight found in the work. It is proved that the operator of a boundary value problem of heat conductivity in an infinite angular domain in a class of growing functions is Noetherian with an index which is equal to minus one. MSC: 35D05, 35K20, 45D05.

An energy space finite element approach for elliptic Dirichlet boundary control problems

In this paper we present a finite element analysis for a Dirichlet boundary control problem where the Dirichlet control is considered in a convex closed subspace of the energy space $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . As an equivalent norm in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) we use the energy norm induced by the so-called Steklov---Poincare operator which realizes the Dirichlet to Neumann map, and which can be implemented by using standard finite element methods. The presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems, and their finite element approximations. Some numerical results are given, which confirm on one hand the theoretical estimates, but on the other hand indicate the differences when modelling the control in $$L_2(\Gamma )$$ L 2 ( Γ ) .

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.

On optimality conditions for optimal control problem in coefficients for Δ p -Laplacian

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as -coefficient of -Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincare Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair. MSC:35J70, 49J20, 49J45, 93C73.

Parametric oscillations and the stability of a mechanical system with considerable dissipation

A detailed investigation is carried out into the problem of parametric oscillations when there is linear dissipation. Using constructive numerical-analytical methods, the boundaries of the domains of stability are constructed for a wide range of variation of the parameters, that is, the modulation factor and the friction coefficient. By solving non-self-adjoint eigenvalue and eigenfunction problems, the phase vectors of the three lower oscillation modes are determined and the principal features of the behaviour of the boundaries when the linear friction coefficient is varied are established. The eigenvalues and eigenfunctions of the adjoint boundary value problem are found. A complete biorthogonal system is constructed and its functional properties are determined. Modified expressions are obtained for scalar products and the squares of the norms of the characteristic phase vectors.

Error analysis of the H1 gradient method for shape-optimization problems of continua

We present an error estimation for the H1 gradient method, which provides numerical solutions to the shape-optimization problem of the domain in which a boundary value problem is dened. The main result is that if second-order elements are used for the solutions of the main and adjoint boundary value problems to evaluate the shape derivative, and the rstorder elements are used for the solution of domain variation in the boundary value problem of the H1 gradient method, then we obtain rst-order convergence of the solution of the domain variation with respect to the size of the nite elements.

Boundary element methods for Dirichlet boundary control problems

AbstractIn this article we discuss the application of boundary element methods for the solution of Dirichlet boundary control problems subject to the Poisson equation with box constraints on the control. The solutions of both the primal and adjoint boundary value problems are given by representation formulae, where the state enters the adjoint problem as volume density. To avoid the related volume potential we apply integration by parts to the representation formula of the adjoint problem. This results in a system of boundary integral equations which is related to the Bi‐Laplacian. For the related Dirichlet to Neumann map, we analyse two different boundary integral representations. The first one is based on the use of single and double layer potentials only, but requires some additional assumptions to ensure stability of the discrete scheme. As a second approach, we consider the symmetric formulation which is based on the use of the Calderon projector and which is stable for standard boundary element discretizations. For both methods, we prove stability and related error estimates which are confirmed by numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

The inverse medium problem in heterogeneous PML-truncated domains using scalar probing waves

We discuss the inverse medium problem associated with semi-infinite domains. In particular, we attempt to image the spatial variability of shear moduli or shear wave velocities from scant surficial measurements of an arbitrarily heterogeneous semi-infinite domain’s response to prescribed dynamic excitations. We use a full waveform approach to drive the inversion process, within a PDE-constrained optimization framework. Due to the semi-infinite extent of the targeted domains, we introduce perfectly-matched-layers (PMLs) to arrive at finite computational domains. The numerical implementation is based on a mixed finite-element method that is used to resolve the ensuing state and adjoint boundary-value problems, both of which are PML-endowed. To alleviate the inherent solution multiplicity, we use Tikhonov and total variation (TV) regularization schemes, in conjunction with a regularization factor continuation scheme. To further improve the optimizer’s chances to converge, we also discuss a source-frequency continuation scheme. We report on two-dimensional numerical experiments using synthetic data. Included are layered profiles, and profiles involving inclined layers and inclusions. We also report on our methodology’s reconstruction of the highly-heterogeneous Marmousi benchmark velocity model.