The numerical method is proposed in this paper to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients and the time-dependent matrices are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities of the involved functions, a methodology for not equally partitioning the time interval is proposed. The main issue of this paper is to obtain an analytic formula of the error bound, where the strong duality theorem for the primal and dual pair of continuous-time linear programming problems with time-dependent matrices and piecewise continuous functions is a by-product. We shall propose two kinds of computational procedure to evaluate the error bounds. One needs to solve the dual problem of the discretized linear programming problem, and another one does not need to solve the dual problem. The detailed differences between these two computational procedures will be also presented. Finally we present a numerical example to demonstrate the usefulness of the numerical method.

中文翻译：

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求解时变矩阵和分段连续函数的连续时间线性规划问题的数值方法

本文提出了一种数值方法来解决一类一般的连续时间线性规划问题，在该问题中，系数中出现的函数和与时间有关的矩阵被假定为分段连续的。为了确保时间间隔的所有子间隔都不会包含所涉及函数的不连续性，提出了一种不等分时间间隔的方法。本文的主要问题是获得误差界的解析公式，其中带有时变矩阵和分段连续函数的原始和对偶连续时间线性规划问题的强对偶定理是副产品。我们将提出两种计算程序来评估误差范围。一个需要解决离散线性规划问题的对偶问题，而另一个则不需要解决对偶问题。还将介绍这两种计算过程之间的详细区别。最后，我们提供一个数值示例，以证明数值方法的有效性。