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Optimality conditions for constrained convex parabolic control problems via duality. (English) Zbl 0963.49017

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 \(L(x, y^*).\)

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
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