A virtual control concept for state constrained optimal control problems

where Ω ⊂ RN , N = 2, 3 is a bounded domain with C0,1-boundary, ν > 0 is a fixed number, yd and yc are given functions from L2(Ω). Furthermore, ua and ub are given real numbers with ua < ub. The difficulty of low regularity of solutions of problems with pointwise state constraints is pointed out in (1). Therefore, different regularization concepts are developed, see e.g. (2), (3), (4) and (5). However, a direct extension of the Lavrentiev regularization concept ((2), (4), (5)) is not possible since the control acts at the boundary. In a recent paper of Tröltzsch and Yousept (6) a source representation was used to overcome this problem. We consider a modified optimal control problem with regularized state constraints by introducing a virtual control v: