On a generalised \(\varepsilon\)-control problem in Banach space (Q1770152)

Let \(X, Y, Z\) be Banach spaces, \(\Omega\) a closed convex subset of \(X\), containing the origin in its interior, \(S:X \rightarrow Y\), \(T:X \rightarrow Z\) linear continuous maps, \(J:X \times Z \rightarrow [0,+\infty[\) a continuous convex function such that \(J(0,0)=0\), \(J(x,z) \rightarrow +\infty\) as \((x,z) \rightarrow \infty\). If \(\phi \in X'\) and \(K\) is a convex subset of \(X\), we denote \([\phi:K]=\{x \in X: \phi (x) = \sup\{\phi (u): u\in K \}\}\). Let \(\varepsilon >0\), \(\alpha \geq 0\), \(\xi \in Y\), \(\eta \in Z\) be such that there exists \[ (\kappa,\zeta) \in A(\alpha)\equiv \{(x,z) \in \Omega \times Z,\;J(x,z) \leq \alpha\} \] with \(| \xi - S \kappa| <\varepsilon\). We suppose either that \(X\) and \(Z\) are reflexive spaces or that there exist normed linear spaces \(X_1\), \(Y_1\), \(Z_1\) and linear maps \(S_1\), \(T_1\) such that \(X=X_1'\), \(Y=Y_1'\), \(Z=Z_1'\), \(S=S_1'\), \(T=T_1'\). Then there exist \(x_0 \in \Omega\) which minimizes \(x \in \{u \in \Omega : | \xi - Su| \leq \varepsilon\} \mapsto J(x,\eta - Tx) \) and \(z_0 \in Z\) such that \((x_0,z_0) \in [(S' \phi_1 + T' \phi_2, \phi_2) : A(\alpha)]\), where \(\alpha\) and \((\phi_1,\phi_2)\) of norm one satisfy each of the following equivalent conditions: \[ \phi_1(\xi)+\phi_2(\eta)=\sup(S'\phi_1+T'\phi_2, \phi_2)(A(\alpha)) +\varepsilon \sup \phi_1(B_Y(0,1)), \tag{a} \] \[ \xi \in Sx_0+\varepsilon [\phi:B_Y(0,1)],\quad \eta=z_0 + Tx_0,\tag{b} \] (c) \(\max \{\psi_1(\xi) + \psi_2(\eta) + \min\{\min\{\min\{J(u,z) - (S'\psi_1 +T'\psi_2)(u) - \psi_2(z) - \varepsilon \psi_1 (y): y \in B_Y(0,1)\}: z \in Z\}: u \in \Omega\}: (\psi_1, \psi_2) \in (Y \times Z)'\} =\phi_1(\xi) +\phi_2(\eta) + \min \{\min\{\min\{J(u,z) - (S'\psi_1 +T'\psi_2)(u) - \psi_2(z) - \varepsilon \psi_1 (y): y \in B_Y(0,1)\}: z \in Z\}: u \in \Omega \}=\alpha\).