Optimal control of parabolic variational inequality involving state constraint (Q1582398)

Necessary conditions for optimality of a pair \((y^*,u^*)\) with respect to a cost functional \[ L(y, u)= \int_0^T [g(t, y(t))+ h(u(t))] dt, \] subject to \(y'+ Ay+ \beta y= Bu\), a.e. in \(Q= \Omega\times (0,T)\), \(y(0)= y_0\), are given in terms of the subdifferentials of the lower-semicontinuous convex function \(h\) and generalized gradients of the locally Lipschitz function \(g\). Many existing results on this problem are extended.