换热器是许多工业产品、工业生产过程以及工程实验中的一种通用装置，用来实现热量从热流体到冷流体的传递以满足规定的工艺要求. 本项目研究与管式换热器方程状态空间解法有关的几个数学问题，包括幂零半群的有界扰动、半群生成元的乘法扰动所产生的发展族以及发展族的幂零性. 幂零性将被用来研究系统的稳恒状态以及达到稳恒状态的时间. 我们关注流体速度随时间变化的一般情形，此时换热器方程可改写成状态空间中非自治的Cauchy问题，其发展族与生成元的乘法扰动有关. 本项目研究的问题具有工程背景，对这些问题的解决不仅具有理论意义也具有实际意义.

As important components of numerous thermal systems and machines, heat exchangers are widely used in a variety of industrial products, processes and engineering experiments to realize the transfer of heat between hot and cold fluids or gases. In this project, we will study some problems which arise when solving the tubular heat exchanger equations using the state space method. More precisely, we will be concerned with bounded perturbations of nilpotent semigroups, evolution families resulting from multiplicative perturbations of semigroup generators and nilpotency of evolution families. The nilpotency will be used to discuss the steady state of the heat exchanger system. As compared with existing researches, we will discuss the general case of time-varying velocities. Accordingly, those corresponding heat exchanger equations should be reformulated into non-autonomous Cauchy problems whose underlying evolution families come from multiplicative perturbations of the semigroup generators. The problems in consideration are practical and the project is meaningful, from both the mathematical and the engineering point of view.