On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem

Abstract

In the paper, a Lagrange optimal control problem governed by a fractional Dirichlet problem with the Riemann–Liouville derivative is considered. To begin with, based on some variational method, the existence and continuous dependence of solution to the aforementioned Dirichlet problem is investigated. Then, continuous dependence is applied to show the existence of optimal solution to the Lagrange problem. An important point is that the solution to Dirichlet problem does need to be unique; therefore, the above dependence should be understood as a continuity of some multifunction—the concept of the Kuratowski–Painlevé limit of the sequence of sets is used to formulate this property.