On Existence and Attainability of Solutions to Optimal Control Problems in Coefficients for Degenerate Variational Inequalities of Monotone Type

Стійкість До Збурень Області Задач Оптимального Керування В Коефіцієнтах Змішаних Систем Типу Гаммерштейна

“…be any τ -convergent sequence of admissible quadruples to the optimal control problem (3.3)(3.7), and let (U 0 , v 0 , y 0 , z 0 ) be its τ -limit. Since the controls {U k } k∈N belong to the set of solenoidal matrices U sol (see (3.2)), it follows from [18,20] that U 0 ∈ U ad and y 0 = y(U 0 ). It remains to show that z 0 ∈ H(v 0 , y 0 ).…”

“…To prove the equalities (4.8)(4.9), we pass to the limit in the integral identities (4.11)(4.12) as ε → 0. Using the arguments from [18,20] and Theorem 3.1, we…”

“…It turns out that this circumstance is the characteristic feature for the majority of optimal control problems in coefcients. To overcome this diculty, in present article, by analogy with [10,18,20], we put some additional constrains on the set of admissible controls. Namely, we consider the matrix-valued controls from the so-called generalized solenoidal set.…”

An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

“…Several results for optimal control problems related to elliptic PDE's with unbounded coefficients have been also obtained in [26,21,22,27,28,9,10] and the recent papers [20], [30] with the reference therein).…”

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

“…Since the range of such optimal control problems is very wide, including as well optimal shape design problems, some problems originating in mechanics and others, this topic has been widely studied by many authors. However, most of these results and methods rely on linear PDEs with bounded coecients in the main part of elliptic operators, while only a few articles deal with with unbounded and degenerate coecients, see [1,3,711,13,14].…”