Sign-Definiteness of the Integral Cost on a System of Linear Integral Equations of Volterra Type

摘要

We study a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to a nondegenerate one, and hence, to obtain new necessary conditions of its nonnegativity. Here we reduce the original problem to a form which differs from the "classical" one only by parameter which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, a new class of problems with reduced cost is investigated and corresponding necessary optimality conditions are obtained.