中文摘要：

混合系统是有交往的分离计算和连续物理过程的动态系统，它变得在我们的现代生活更普通、更不可缺少、更复杂。特别地，他们中的许多是安全批评的，因此被要求满足一个批评安全标准。不变的产生在混合系统的确认和合成起一个中央作用。在以前的工作，第四个作者和他的合作者为是的一个半代数学的集合给了一个必要、足够的条件一一个多项式不变自治动态系统，它把一个确定的答案给了开的问题。另外，基于哪个为产生一个给定的多项式的所有半代数学的 invariants 的一个完全的算法有给定的形状的自治混合系统被建议。这份报纸考虑怎么扩大他们的工作到非自治的动态、混合的系统。非自治的动态、混合的系统与输入，它在实践是很普通的；相反，自治的没有输入。而且，在场的作者为非自治的多项式混血儿系统验证半代数学的 invariants 的一个健全、完全的算法。基于哪个，作者建议一个声音和完全的算法与一个预定义的模板产生所有 invariants。

英文摘要：

Hybrid systems are dynamical systems with interacting discrete computation and contin- uous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template.