REGULARITY OF LAGRANGE MULTIPLIERS FOR OPTIMAL CONTROL PROBLEMS WITH PDEs AND MIXED CONTROL STATE CONSTRAINTS

Lagrange multipliers for distributed parameter systems with mixed control-state constraints may exhibit better regularity properties than those for problems with pure pointwise state constraints, (1), (2), (4). Under natural assumptions, they are functions of certain L-spaces, while Lagrange multpliers for pointwise state constraints are, in general, measures. Following an approach suggested in (3) for ODEs, a new and simplified technique is applied to prove L-regularity in the case of elliptic PDEs. Moreover, an idea of (5) is extended to derive L-estimates for the Lagrange multipliers, along with the proof of Lipschitz regularity of optimal controls.