General Optimality Conditions for Constrained Convex Control Problems

Optimal control of variational inequalities: A mathematical programming approach

“…Similar problems have been studied also in Bergounioux and Tiba (1994) when the set 1J is convex. Here, the main difficulty comes from the fact that the feasible domain 1J is not convex because of the bilinear constraint "(y, {) = 0".…”

“…Here, the main difficulty comes from the fact that the feasible domain 1J is not convex because of the bilinear constraint "(y, {) = 0". So, we cannot use directly the convex analysis methods that have been used for instance in Bergounioux and Tiba (1994).…”

The control variational method for beams in contact with deformable obstacles

“…Note that the function r * ∈ L 2 (0, L) is the Lagrange multiplier associated to the state equation (4.3). Similar arguments have been used in [2] in order to derive first order optimality conditions in the study of abstract control problems for evolution equations. Note also that conditions (4.17) or (4.19) express the fact that the gradient of the minimized functional (4.5) (or (4.2)), is zero (or has a certain orientation with respect to the constraints) at the minimizer.…”

Optimal control of a semilinear parabolic equation with singular arcs

“…When the control is distributed, if the control is out of bounds on some open set, it is sometimes possible to give an explicit expression of the control: this is the theory of generalized bang-bang control, see Bergounioux and Tiba [3], Tröltzsch [27], Bonnans and Tiba [8].…”