We investigate the solution and stability of continuous-time cross-dimensional linear systems (CCDLSs) with dimension bounded by V-addition and V-product. Using the integral iteration method, the solution to CCDLSs can be obtained. Based on the new algebraic expression of the solution and the Jordan decomposition method of matrix, a necessary and sufficient condition is derived for judging whether a CCDLS is asymptotically stable with a given initial state. This condition demonstrates a method for finding the domain of attraction and its relationships. Then, all the initial states that can be stabilized are studied, and a method for designing the corresponding controller is proposed. Two examples are presented to illustrate the validity of the theoretical results.

我们研究了以V加法和V乘积为界的连续时间跨维线性系统（CCDLSs）的解和稳定性。使用积分迭代法，可以获得CCDLS的解。基于该解的新代数表达式和矩阵的Jordan分解方法，得出判断CCDLS在给定初始状态下是否渐近稳定的充要条件。此条件演示了一种寻找吸引力域及其关系的方法。然后，研究了所有可以稳定的初始状态，并提出了一种设计相应控制器的方法。给出两个例子来说明理论结果的有效性。