49 research outputs found
Approximation of an Optimal Control Problem in the Coefficient for Variational Inequality with Anisotropic p-Laplacian
Get PDFWe study an optimal control
problem for a variational inequality with the so-called anisotropic -Laplacian in the principle part of this inequality. The coefficients of the anisotropic -Laplacian, the matrix , we take as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the solutions of the corresponding variational inequality. We show that the original problem is well-posed and derive existence of optimal pairs. Since the anisotropic -Laplacian inherits the degeneracy with respect to unboundedness of the term , we introduce a two-parameter model for the relaxation of the original problem. Further we discuss the asymptotic behavior of relaxed solutions and show that some optimal pairs to the original problem can be attained by the solutions of two-parametric approximated optimal control problems
Analysis of an optimal control problem for the case of a strongly degenerate elliptic equation.
No full textOn Optimal Controls in Coefficients for Ill-Posed Non-Linear Elliptic Dirichlet Boundary Value Problems
Get PDFWe consider an optimal control problem (OCP) associated to Dirichlet boundary value problem for non-linear elliptic
equation on a bounded domain . We take the
coefficient in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix . We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-posed and
solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices . We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP
On Optimal Boundary Control Problem for a Strongly Degenerate Elliptic Equation
No full textIn this paper we study an optimal control problem for a linear boundary value problem with strongly degenerate coefficient in the main part of the elliptic operator and with the boundary control. Given a cost functional, the objective is to provide the well-posedness analysis of the corresponding optimal control problem, prove existence of the optimal solutions and propose the scheme for their approximation
On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality
Get PDFWe consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions
Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type
No full textIn this paper we consider an optimal control problem (OCP) for
the coupled system of a nonlinear monotone Dirichlet problem with matrixvalued
L^Infinity(Ω;R^NxN)-controls in coefficients and a nonlinear equation of Hammerstein
type. Since problems of this type have no solutions in general, we
make a special assumption on the coefficients of the state equation and introduce
the class of so-called solenoidal admissible controls. Using the direct
method in calculus of variations, we prove the existence of an optimal control.
We also study the stability of the optimal control problem with respect to the
domain perturbation. In particular, we derive the sufficient conditions of the
Mosco-stability for the given class of OCPs
On optimality conditions for optimal control problem in coefficients for Delta_p-Laplacian
Get PDFIn this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as L^infinity(Omega) coefficient of Delta_p-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-Poincaré Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair
On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations
No full textWe discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation
On a Variational Problem with Nonstandard Growth Functional and its Applications to Image Processing
No full textWe propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations
On the Existence of Weak Optimal Controls in the Coefficients for a Degenerate Anisotropic p-Laplacian
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