Distributed optimal control problems for a class of elliptic hemivariational inequalities with a parameter and its asymptotic behavior

Simultaneous distributed and Neumann boundary optimal control problems for elliptic hemivariational inequalities

“…We know, by [30], that there exists a unique optimal pair (g, q) ∈ H × Q of the simultaneous distributed-boundary optimal control problem (1.8). In similar way to [12], we have a result on existence of solution to the simultaneous optimal control problem (1.10) in which the system is governed by the hemivariational inequality (1.4). Theorem 4.1.…”

“…Subsequently, in a similar way to [12,26] (see more details in Theorem 5.1, Step 2), from (3.3), by the compactness of the trace operator, the upper semicontinuity of the function R × R (r, s) → j 0 (x, r; s) ∈ R for a.e. x ∈ Γ 3 and H( j)(d), we prove that j 0 (x, η; b − η) = 0 a.e.…”

“…where u q is the unique solution to the variational equality (1.2), u αq is a solution to the hemivariational inequality (1.4), z d ∈ H given, and M 2 a positive constant. This complement to the one studied in [12,13,29]. • A simultaneous distributed and Neumann boundary optimal control problem, given by [13,28,30]:…”

Numerical analysis of a family of simultaneous distributed-boundary mixed elliptic optimal control problems and their asymptotic behaviour through a commutative diagram and error estimates

“…with c different positive constants independent of h. Moreover, by using the variational equality (17), we obtain…”

“…By using the variational equality (18), we can pass to the limit when α → +∞, and by uniqueness of the variational equality (17) we obtain that η hgq = u hgq . By using the above properties, and the variational equalities ( 17) and ( 18), we deduce (36) and by using a similar method we can obtain the limit (37) for the discrete adjoint system state.…”