Degenerate problems of optimal control. I

Considered is the practically important and theoretically challenging class of optimal control problems which can be integrated within the common notion of ‘degenerate problems.’ The general definition of such problems is given, which arises from the connection between the degeneracy and the presence of hidden passive differential constraints or discrete chains in the problem. This definition is analyzed with the focus on its relation with the classical notion of degeneracy in the variational calculus and the notion of singular and sliding modes well known in the control theory. This paper is the first one in the series of three, which are aimed at presenting a survey of the main facts and applications of the special theory of such problems, which is essentially based on finding and eliminating passive constraints. New results and generalizations are also reported.