Vol. 33, No. 1, 1970

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Trajectory integrals of set valued functions

T. F. Bridgland

Vol. 33 (1970), No. 1, 43–68

Let I be a compact interval of the real line and for each t in I, let F(t) denote a nonvoid subset of euclidean n-space En. Let I(F) be the collection of all Lebesgue summable functions u;I En having the property that u(t) F(t) almost everywhere on I. Following the lead of Kudo and Richter, Aumann defines the integral of F over I by

∫          ∫
F(t)dt = { f(t)dt | f ∈ ℱ (F )}
I          I            I

and, in addition to other results, establishes a dominated convergence theorem for such integrals. Hermes has pursued Aumann’s line of thought to obtain results concerning something akin to a “derivative” for set valued functions.

It is certainly also valid (and for control theoretic applications essential) to define the trajectory integral of F to be the set 𝒮I(F) of all functions which vanish at the left endpoint of I and have derivatives in I(F). The purpose of this paper is taken to be the study of the trajectory integrals of nonvoid, compact set valued functions. A primary goal is the extension of the results of Aumann to include the trajectory integral. A secondary goal is the provision of an intuitively meaningful definition of “derivative” for set valued functions.

Mathematical Subject Classification
Primary: 28.25
Received: 3 January 1969
Revised: 11 September 1969
Published: 1 April 1970
T. F. Bridgland