On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Abstract

We 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 a Nonstandard Growth Functional and Its Applications to Image Processing

“…, M} be a given spectral channel. Mainly basing on the concept of relaxation of extremal problems and their variational convergence [31,33,38], we propose the following algorithm. At the first step, we set up…”

“…Since the uniqueness of solutions to ( 16) is a rather questionable option, it follows that, in principle, these definitions can describe different functions in i . As immediately follows from (38), a weak solution is a merely feasible one to the original problem. However, if the problem ( 16) admits a unique solution (u 0 i , p 0 i ) ∈ i , then (38) implies that this function is considered as a weak solution.…”

“…As immediately follows from (38), a weak solution is a merely feasible one to the original problem. However, if the problem ( 16) admits a unique solution (u 0 i , p 0 i ) ∈ i , then (38) implies that this function is considered as a weak solution. On the other hand, if u * i ∈ i is some solution to (16), then setting in (34)…”

“…To do so, we notice that if v ∈ i is a feasible solution to (16), then F(v(•)) is subjected to the two-side inequality (28) with δ ∈ (0, 1) given by (29). Keeping this in mind and following in some aspects the standard penalty method (see, [52,Chapter 2], see also [34][35][36][37]40]), we consider the following family of approximating problems:…”