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

Abstract

LetH1 andH2 be Hilbert spaces and letN be an algebraic subspace ofH1. The least-squares problem for a linear relationLźH1źH2 restricted to an algebraic cosetS:=g+N, g ź H1, is considered. Various characterizations of a minimizer are derived in the form of inclusion relations that incorporate the constraints, and conditions under which a minimizer exists are developed. In particular, it is shown that generalized forms of the "normal equations" for constrained least-squares problems 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. An application of the abstract theory is given to a singular control problem involving ordinary differential equations with generalized boundary conditions, where the control may generate multiresponses. Characterizations of an optimal solution are developed in the form of inclusions that involve either an integrodifferential equation or a differential equation, and adjoint subspaces and/or solutions of certain linear equations.