A remark on the Kochen-Specker theorem and some characterizations of the determinant on sets of Hermitian matrices

In this paper we describe the general forms of all (nonlinear) continuous functionals on the sets of positive definite, positive semi-definite and Hermitian matrices which are multiplicative on the commuting elements. As a consequence, we obtain some new characterizations of the determinant on those classes of matrices.

     

Optimal control of a system governed by a parabolic equation with an infinite number of variables

A distributed control problem for a parabolic operator with an infinite number of variables is considered. The performance index is more general than the quadratic one an has an integral form. Making use of the Dubovicki-Milutin theorem, necessary and sufficient conditions of optimality are derived for the Dirichlet problem.     

Optimal control of a system governed by a parabolic equation with an infinite number of variables and time delay

A distributed control problem with delay for the parabolic operator with an infinite number of variables is considered. The performance index has an integral form. Constraints on controls are assumed. To obtain optimality conditions, the generalization of the Dubovicki-Milutin theorem given by Walczak in Ref. 1 was applied.