This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability of the control-to-state mapping. The paper ends with the application of the general results to a semilinear heat equation involving the max-function.
The paper deals with a viscous damage model including two damage variables, a local and a non-local one, which are coupled through a penalty term in the free energy functional. Under certain regularity conditions for linear elasticity equations, existence and uniqueness of the solution is proven, provided that the penalization parameter is chosen sufficiently large. Moreover, the regularity of the unique solution is investigated, in particular the differentiability w.r.t. time.
The paper deals with a class of optimal control problems governed by a viscous damage model including two damage variables which are coupled through a penalty term. The existence and directional differentiability of the associated control‐to‐state operator is established. Moreover, necessary optimality conditions are derived. It is shown that these are equivalent to an optimality system, provided that a strict complementarity condition is satisfied.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.