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

Abstract

We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain Ω. We take the matrixvalued coecients A(x) of such system as a control in L 1 (Ω; R N × R N ). One of the important features of the admissible controls is the fact that the coecient matrices A(x) are non-symmetric, unbounded on Ω, and eigenvalues of the symmetric part A sym = (A + A t )/2 may vanish in Ω.

On optimal <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-control in coefficients for quasi-linear Dirichlet boundary value problems with <inline-formula><tex-math id="M2">\begin{document}$ BMO $\end{document}</tex-math></inline-formula>-anisotropic <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian

“…, we see that conditions (i)-(ii) and estimates (15)- (16) immediately imply the boundedness of this sequence in W 1,1 (Ω; M N ) and in variable spaces H 1,p 0,B k (Ω) and L p (Ω, u k dx). Moreover, by inequalities (19)-(20), we have the compact embedding…”

“…or apply the procedure of the direct Steklov smoothing to a given matrix A ∈ M ad (Ω) with some positive compactly supported smooth kernel (see, for instance, [16]). (5)), it follows that the sequence L −1 k k∈N is equiintegrable.…”

“…In particular, the unboundedness of the skew-symmetric part of A can have a reflection in non-uniqueness of weak solutions to the corresponding (BVP). For more details and other notions of solutions to elliptic equations with unbounded coefficients we refer to [8,15,16,18,37]. So, in contrast to the paper [39], where the author consider the case of well-posed Dirichlet boundary value problem for a quasi-linear elliptic equation with unbounded coefficients in its principal part, we deal with the optimal control problem for ill-posed control object.…”

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

“…is the variational limit of sequence (8) in the sense of Definition 1.2 and this problem has a nonempty set of solutions…”

“…There is another type of weak solutions called non-variational [20,22], singular [3,13,14,19], pathological [16,17] and others. As for the optimal control problem (1) we have the following result [9] (see [8] for comparison): for any approximation {A * k } k∈N of the matrix A * ∈ L 2 Ω; S N skew with properties {A * k } k∈N ⊂ L ∞ (Ω; S N skew ) and A * k → A * strongly in L 2 (Ω; S N skew ), optimal solutions to the corresponding regularized OCPs associated with matrices A * k always lead in the limit as k → ∞ to some admissible (but not optimal in general) solution ( A, y ) of the original OCP (1). Moreover, this limit pair can depend on the choice of the approximative sequence {A * k } k∈N .…”