JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 24, 608-612 (1968) On the Convergence of an improper Integral Evaluated along the Solution of a Differential Equation* YJ. H. MCCLAMROCH Computer, Information and Control Engineering, The University of Michigan, Ann Arbor, Michigan 48104 AND J. K. AGGARWAL Department of Electrical Engineering, The University of Texas, Austin, Texas 78712 INTRODUCTION The study of the convergence and the evaluation of improper integrals has received much attention. In this paper, we shall consider conditions under which the convergence of a particular class of improper integrals can be guaranteed. In particular, the integrand is not specified in closed form but is evaluated in terms of the unique solution to an associated differential equation. The improper integral can be written as where g(x, t) is a specified scalar function of the n-vector x and the time t. The vector function +(t, x0 , to) represents the solution to the vector differ- ential equation dx x E R =f(x, t) with the initial condition x(t,) = x0 . (3) f (x, t) is a function with values in the Euclidean space Rn which is defined on some set S x 1 = {(x, t) E Rn x R 1 11 x /I < r, t > O}. It is assumed that * This research was partially supported by the United States Air Force under Grant No. AF-AFOSR-814-66 and by the National Science Foundation under Grant No. GK-1879. 608