Constrained least-squares solutions of linear inclusions and singular control problems in Hilbert spaces (Q1117443)

The least square problem for a linear relation \(L\subset H_ 1\oplus H_ 2\), where \(H_ 1\) and \(H_ 2\) are Hilbert spaces and N is an algebraic subspace of \(H_ 1\), restricted to an algebraic coset S: g\(+N\), \(g\in H_ 1\), is considered. Various characterizations of a minimizer are derived in the form of inclusion relations that incorporate the constraints. It is shown that generalized forms of the ``normal equations'' for constrained least-square problem become ``normal inclusions'' that involve a multivalued adjoint. The setting includes large classes of constrained least-squares minimization problems and optimal control problems subject to generalized boundary conditions. The theory is applied to a singular control problem involving ordinary differential equations with generalized boundary conditions, where control may generate multiresponses. Characterizations of an optimal solution are developed in the form of inclusions that involve either an integro-differential or a differential equation, an adjoint subspaces and/or solutions of some linear equations.