Azé, D.; Bolintinéanu, S. Optimality conditions for constrained convex parabolic control problems via duality. (English) Zbl 0963.49017 J. Convex Anal. 7, No. 1, 1-17 (2000). The problem is that of minimizing \[ J(y, u) = \int_0^T L(t, y(t), u(t)) dt + l(y(0), y(T)) \] subject to \[ y'(t) + A(t)y(t) = B(t)u(t) + f(t), \quad 0 \leq t \leq T, \]\[ (y, u) \in {\mathcal M} \subset E(0, T) \times L^p(0, T; U), \] where \(E(0, T)\) is a space of functions defined in \(0 \leq t \leq T.\) The abstract framework used by the authors is that of duality in Fréchet spaces. Given a proper, convex lower semicontinuous function \(F : X \times Y \to R \cup \{+\infty\}\) the primal and dual problems are, respectively, \[ \inf_{x \in X} F(x, 0) \quad \text{and} \quad \sup_{y^* \in Y^*} (-F^*(0, y^*)) \] with \(F^*\) the conjugate of \(F,\) and the solutions \(x_0 \in X\) of the primal and \(y^*_0 \in Y\) of the dual are characterized by \((x_0, y^*_0)\) being a saddle point of the Lagrangian \(L(x, y^*).\) Reviewer: Hector O.Fattorini (Los Angeles) Cited in 6 Documents MSC: 49K20 Optimality conditions for problems involving partial differential equations 49N15 Duality theory (optimization) 49J20 Existence theories for optimal control problems involving partial differential equations 49J27 Existence theories for problems in abstract spaces 49K27 Optimality conditions for problems in abstract spaces Keywords:constrained optimization; convex control problems; parabolic control problems; duality PDFBibTeX XMLCite \textit{D. Azé} and \textit{S. Bolintinéanu}, J. Convex Anal. 7, No. 1, 1--17 (2000; Zbl 0963.49017) Full Text: EuDML EMIS