Optimal control of a class of nonlinear equations in Banach space (Q748865)

Let \({\mathbb{X}}\) be a real reflexive Banach space. The author considers the minimization problem minimize \(l_ 0(u)\) subject to \(u\in U_{ad}\), where \(l_ 0(u)=\| Cx(u)-w_ 0\|^ 2_ W+\mu \| u\|^ 2_ U\), \(U_{ad}\) is a closed convex set in a reflexive Banach space U, C is a linear operator from \({\mathbb{X}}\) into a Banach space W and \(x=x(u)\) is a solution of the Hammerstein type equation \(x+F_ 2F_ 1(x)=f_ 0+Bu\) where the nonlinear operators \(F_ j\) and the linear operator B satisfy suitable conditions. This paper presents several results (including existence) on this problem.