Integral state-constrained optimal control problems for some quasilinear parabolic equations (Q1968743)

The paper considers problems of minimizing integral functionals (depending on state and controls) on triples \((y,u,v)\) satisfying \[ y_t= \text{div }a(x,t,y,\nabla y,u)- a_0(x, t,y,\nabla y,u)\quad\text{in }Q= \Omega\times (0,T), \] \[ y(x,0)= v\quad\text{in }\Omega,\quad y(x,t)= 0\quad\text{on }\partial\Omega\times (0,T), \] \[ I_i(y)= 0,\quad i= 1,\dots, m_1,\quad J_j(y)\leq 0,\quad j= 1,\dots, m_2,\quad (u,v)\in K. \] Here \(\Omega\subset\mathbb{R}^n\) is a bounded Lipschitz domain, \(y\) is the state, \((u,v)\) is the pair of controls from a closed convex set \(K\subset L_2(Q)\times L_2(\Omega)\) and \(I_i\), \(J_j\) are integral functionals. The author shows that under some appropriate conditions (general enough) on the functions \(a= (a_1,\dots, a_n)\), \(a_0\) and the integrands of functionals, the relation control -- state and the functionals are continuous and Gâteaux differentiable. After that necessary optimality conditions in the standard Lagrange form are obtained.