Appl Math Optim 36:243–262 (1997)
1997 Springer-Verlag New York Inc.
Quadratic Functionals with General Boundary Conditions
Z. Doˇsl´a and O. Doˇsl´y
Department of Mathematics, Masaryk University,
Jan´aˇckovo n´am. 2a, 66295 Brno, Czech Republic
Abstract. The purpose of this paper is to give the Reid “Roundabout Theorem” for
quadratic functionals with general boundary conditions. In particular, we describe
the so-called coupled point and regularity condition introduced in  in terms of
Riccati equation solutions.
Key Words. Quadratic functional, Coupled point, Regularity condition, Riccati
equation, Picone’s identity.
AMS Classiﬁcation. 49B10, 34C10.
In this paper we continue the study of positivity of the quadratic functional with general
boundary conditions that was began in  and developed in various directions in ,
, , , and . Here, as in , we consider the quadratic functional
J (η) =
(t)P(t)η(t) + 2η
(t)Q(t )η(t) + η
(t)} dt (1.1)
over the class of absolutely continuous functions η such that η
(a, b), subject to
the boundary conditions
= 0. (1.2)
Such functions are said to be admissible for (1.1). Here P, Q, R are n × n matrix-valued
functions, D, are 2n × 2n constant matrixes, P ∈ L
(a, b), Q ∈ L
(a, b), R is