Sequential extension in the problem of control in coefficients for elliptic-type equations

Abstract

| The problem of control in coe± cients for the elliptic-type equation is considered. For this problem, set by Lions [1], it is not too di± cult a matter to obtain necessary optimality conditions under the assumption that the optimal control does exist; yet, solvability of this problem is hard to establish. Nonetheless, it makes sense to search for an admissible control such that the value of the functional calculated on this control is as close as is wished to its lower bound. In this paper, a method for¯nding minimizing sequences is proposed. The approach used is based on sequential extension of the problem which is analogous to the scheme of complement of spaces [2]. For the extended problem, necessary conditions for sequential optimum property, which characterize minimizing sequences, are established. With the example of Varga [3], we show that, following this method, one can¯nd the minimizing sequences even if the optimal control does not exist.

Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls

“…which is closed in the sense of H-convergence; and in [4] for U = {A, B}, where A and B are two given positive definite matrices and f 0 also depends on ∇y (•). See, e.g., [6], [7], [9], [11], [12] for other relevant works.…”

Optimality conditions for semilinear parabolic equations with controls in leading term

“…When A(t, x, u) ≡ A(t, x), Problem (C) has been studied by many authors, see [10] and the references cited therein. Works concerning the elliptic cases with leading term containing controls can be founded in [4][5][6][7][11][12][13][14], etc. However, it seems that there are only few works devoted to parabolic cases (see [3,15], etc.…”