Abstract
In this article, the fractional optimal control problem for variational inequalities is considered. The fractional time derivative is considered in Riemann-Liouville sense. The existence and uniqueness of solutions to a class of fractional differential variational inequalities in a Sobolev space is studied. An application to a fractional variational inequality in a bounded domain with Dirichlet and Neumann boundary conditions is given. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the fractional Cauchy problems with the quadratic performance functional are derived. Specifically, the Euler-Lagrange equations first order optimality condition with an adjoint problem are presented.