Optimal Control Of Coefficients Research Articles

Optimal control of coefficients in parabolic free boundary problems modeling laser ablation

Inverse Stefan type free boundary problem for the second order parabolic equation arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. New PDE constrained optimal control framework in Hilbert–Besov spaces introduced in Abdulla (2013) is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proved. Numerical algorithm based on the Fréchet differentiability and the gradient method in Hilbert–Besov spaces is implemented. The results of the numerical experiments for the identification of the free boundary and diffusion coefficient are presented.

On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$ . We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ 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 $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ . 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-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$ . 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 the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems

We consider the inverse Stefan type free boundary problem, where the coefficients, boundary heat flux, and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue an optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov-Holder spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov-Holder spaces for the numerical solution of the inverse Stefan problem.

On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued \(L^\infty (\varOmega ,\mathbb {R}^{N\times N})\)-controls in its coefficients and a nonlinear equation of Hammerstein type. Using the direct method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization.

On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems

On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems

О дифференцируемости по Фреше функционала качества в задачах оптимального управления коэффициентами эллиптических уравнений

О дифференцируемости по Фреше функционала качества в задачах оптимального управления коэффициентами эллиптических уравнений

Optimal control in coefficients for degenerate linear parabolic equations

The aim of this work is to study the optimal control problem associated to a linear parabolic equation with homogeneous Dirichlet boundary condition. The control variable is the matrix of \(L^1\)-coefficients in the main part of the parabolic operator. The precise answer about existence or none-existence of an \(L^1\)-optimal solution heavily depends on the class of admissible controls. The main questions concern the right setting of the optimal control problem with \(L^1\)-controls in the coefficients, and the right class of admissible solutions to the above problem. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problem in the class of \(H\)-admissible solutions.

On optimal control of the hyperbolic equation coefficients

For the problem of optimal control of the coefficients of the linear hyperbolic equation, consideration was given to the issues of correct formulation, and the necessary optimality condition was established.

Weak optimal controls in coefficients for linear elliptic problems

In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. We adopt the weight function as a control in L 1(Ω). Using the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.

On the optimal control of coefficients in elliptic problems. Application to the optimization of the head slider

We consider an optimal control problem for a class of non-linear elliptic equations. A result of existence and uniqueness of the state equation is proven under weaker hypotheses than in the literature. We also prove the existence of an optimal control. Applications to some lubrication problems and numerical results are given.

Optimal control in coefficients of boundary value problems with a unilateral inner obstacle

We will deal with some classes of nonsmooth optimization problems, where the state problem is given in the form of a state variational inequality with coefficients as the control variable. Existence of a nonsmooth optimal control is proven on the abstract level. Using the results of G-convergence theory it is shown that there exists an optimal solution of an extended control problem.

Optimal control in coefficients of elliptic equations with state constraints

This paper is concerned with state constrained optimal control problems of elliptic equations, the control being a coefficient of the partial differential equation. Existence of an optimal control is proved and optimality conditions are derived. We perform finite-element approximations of optimal control problems and state some convergence results: we prove convergence of optimal controls and states as well as convergence of Lagrange multipliers.

Optimal control in coefficients for weak variational problems in Hilbert space

In this paper we give sufficient conditions for the existence of solutions of a problem of parametric optimization. We use continuity with respect to a functional parameter of weak solutions of a variational problem in a Hilbert space. We consider a problem of optimization with the control in coefficients of linear parabolic equation as an example. Using results of Spagnolo we characterize the closure of the reachable set. Finally, we construct an example of an optimization problem with the control in coefficients of a parabolic equation which does not have an optimal solution.

Optimal control of coefficients in linear elliptic equations i. existence of optimal controls

In the paper the problem of existence of optimal control is considered for a control problem with a linear elliptic equation whose coefficients depend on the control.