本文中我们首先对无约束条件时的有限时间区间上时变系统线性二次型（FHLQ）问题进行了分类，并给出了问题属于每一类的充分必要条件（定理2）．其次证明了对于HLLQ问题其下确界有限与其可达是等价的（定理1）在此基础上给出了问题为正则的一些新的条件（定理3）．最后，我们通过新的途径讨论了有控制能量约束时的HLLQ问题 In this paper, we first classify the linear quadratic system (FHLQ) problems for time-varying systems in finite time intervals under unconstrained conditions and give the sufficient and necessary conditions for the problems to belong to each category (Theorem 2). Second, it is proved that the lower bound of the real solution to the HLLQ problem is equivalent to its reachability (Theorem 1). On this basis, some new conditions for which the problem is regular (Theorem 3) are given. Finally, we discuss the HLLQ problem with control of energy constraints through a new approach