从抽象线性系统的可检测性到指数输出到状态稳定性及其逆问题
From detectability of abstract linear systems to exponential output-to-state stability and back
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中文总结 AI 辅助
本文研究Banach空间中线性无穷维系统的指数输出到状态稳定性（eOSS），证明其与蕴含形式强制性Lyapunov函数等价，并探讨可检测性、零可检测性及有限维不稳定子空间下的充要条件，以抛物方程为例验证。
中文摘要 AI 辅助
我们研究了Banach空间中具有有界输出算子的线性无穷维系统的指数输出到状态稳定性（eOSS）。研究表明，eOSS等价于存在一个蕴含形式的强制性eOSS Lyapunov函数。此外，指数可检测性保证了耗散形式的强制性eOSS Lyapunov函数的存在，从而保证了eOSS。反例表明，逆蕴含关系一般不成立：指数零可检测性不蕴含eOSS，eOSS不蕴含耗散形式的eOSS Lyapunov函数的存在，而后者又不蕴含指数可检测性。如果不稳定子空间是有限维的，则零可检测性蕴含指数可检测性，并给出eOSS的等价刻画。结果通过一个抛物型方程加以说明。
英文摘要
We study exponential output-to-state stability (eOSS) of linear infinite-dimensional systems in Banach spaces with bounded output operators. It is shown that eOSS is equivalent to the existence of a coercive eOSS Lyapunov function in implication form. Also exponential detectability guarantees the existence of a coercive eOSS Lyapunov function in dissipation form and therefore eOSS. Counterexamples demonstrate that the converse implications fail in general: exponential zero-detectability does not imply eOSS, eOSS does not imply the existence of an eOSS Lyapunov function in dissipation form, which, in turn, does not imply exponential detectability. If the unstable subspace is finite-dimensional, zero-detectability implies exponential detectability and yields equivalent characterizations of eOSS. The results are illustrated with a parabolic equation.
发表机构
- Faculty of Computer Science and Mathematics, University of Passau（帕绍大学计算机科学与数学学院）
- Department of Mathematics, University of Bayreuth（拜罗伊特大学数学系）
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