Equivalent conditions to the nonnegativity of a quadratic functional in discrete optimal control

摘要

In this paper we provide a characterization of the nonnegativity of a discrete quadratic functional I with fixed right endpoint in the optimal control setting. This characterization is closely related to the kernel condition earlier introduced by M. Bohner as a part of a focal points definition for conjoined bases of the associated linear Hamiltonian difference system. When this kernel condition is satisfied only up to a certain critical index m, the traditional conditions, which are the focal points, conjugate intervals, implicit Riccati equation, and partial quadratic functionals, must be replaced by a new condition. This new condition is determined to be the nonnegativity of a block tridiagonal matrix, representing the remainder of I after the index m, on a suitable subspace. Applications of our result include the discrete Jacobi condition, a unification of the nonnegativity and positivity of I into one statement, and an improved result for the special case of the discrete calculus of variations. Even when both endpoints of I are fixed, this paper provides a new result.