Optimal control problems with mixed control-state constraints (Q2706156)

The paper studies regularity of Lagrange multipliers for mixed control-state pointwise constraints. The authors consider optimal control problems for the semilinear parabolic equation NEWLINE\[NEWLINEy_t= \text{div}(A(x)\nabla y)+ f(x,t,y)\quad\text{in }\Omega\times (0,T)NEWLINE\]NEWLINE with the boundary condition NEWLINE\[NEWLINE{\partial\over\partial n_A} y= \varphi(x, t,y,v)\quad\text{on }\partial\Omega\times (0,T).NEWLINE\]NEWLINE Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain with \(C^2\) boundary \(\partial\Omega\), \(v\) is the control and \(n_A\) is the conormal. Control-state constraints are of the form NEWLINE\[NEWLINE(g_1(y(\cdot), v(\cdot)),\dots, g_m(y(\cdot), v(\cdot)))\in D\subset L_\infty(\partial\Omega\times (0,T); \mathbb{R}^m),NEWLINE\]NEWLINE where \(D\) is a convex closed set with nonempty interior. For such constraints the standard optimality conditions involve the adjoint equation with a measure in the right-hand side. The authors give sufficient conditions that the right-hand side of the adjoint equation in reality is an element of a Lebesgue space. For instance, it is so if the derivatives \(g_{jv}'(y(\cdot), v(\cdot))\), \(j= 1,\dots, m\), are positive on the optimal pair \((y_0, v_0)\).