Optimal Control of Linear

The regularity of Lagrange multipliers for state-constrained optimal control problems belongs to the basic questions of control theory. Here, we investigate bottleneck problems arising from optimal control problems for PDEs with certain mixed control-state inequality constraints. We show how to obtain Lagrange multipliers in L p-spaces for linear problems and give an application to linear parabolic optimal control problems. 1. Introduction The regularity of Lagrange multipliers for state-constrained optimal control problems belongs to the basic questions of control theory. This is known for control systems governed by ODEs and refers to PDEs as well. In many cases, Lagrange multipliers have to be measures, for instance, if pointwise state-constraints are given in distributed parameter systems. Unfortunately, several methods cannot be applied for measures as Lagrange multipliers. In particular, standard second order suucient optimality conditions cannot be established, and the convergence of some numerical techniques is not ensured. It is well known that Lagrange multipliers can be obtained for nonlinear programming problems in Banach spaces (including control problems) by considering the associated linearized problem (see Zowe and Kurcyusz 15]). In this way, the theory of linear programming problems in Banach spaces may serve as a tool to prove the existence of Lagrange multipliers.