Saddle‐point optimality criteria in modified variational control problems with partial differential equation constraints

Abstract

SummaryIn this article, based on a multidimensional control problem, in short (MCP), we introduce a modified multidimensional variational control problem involving first‐order partial differential equations and inequality‐type constraints. As well, we formulate and prove optimality conditions for this new variational control problem. Furthermore, we establish (under some generalized convexity assumptions) an equivalence between an optimal solution of (MCP) and a saddle‐point associated with the Lagrange functional (Lagrangian) corresponding to the modified multidimensional control problem. Also, in order to illustrate our main characterization results and their effectiveness, we present several applications.