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→控制论中的耗散系统 Dissipative systems in control theory
Willems first introduced the concept of dissipativity in systems theory to describe dynamical systems by input-output properties. Considering a dynamical system described by its state 𝑥(𝑡), its input 𝑢(𝑡) and its output 𝑦(𝑡), the input-output correlation is given a supply rate 𝑤(𝑢(𝑡),𝑦(𝑡)). A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function 𝑉(𝑥(𝑡)) such that 𝑉(0)=0, 𝑉(𝑥(𝑡))≥0 and𝑉˙(𝑥(𝑡))≤𝑤(𝑢(𝑡),𝑦(𝑡)).
Willems first introduced the concept of dissipativity in systems theory to describe dynamical systems by input-output properties. Considering a dynamical system described by its state 𝑥(𝑡), its input 𝑢(𝑡) and its output 𝑦(𝑡), the input-output correlation is given a supply rate 𝑤(𝑢(𝑡),𝑦(𝑡)). A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function 𝑉(𝑥(𝑡)) such that 𝑉(0)=0, 𝑉(𝑥(𝑡))≥0 and𝑉˙(𝑥(𝑡))≤𝑤(𝑢(𝑡),𝑦(𝑡)).
Willems首先在系统理论中引入耗散性的概念，用输入输出特性来描述动力系统。考虑一个由其状态𝑥(𝑡)、输入𝑢(𝑡)和输出𝑦(𝑡)所描述的动力系统，输入-输出的相关性被给出了供给率𝑤(𝑢(𝑡),𝑦(𝑡))。如果存在一个连续可微的存储函数𝑉(𝑥(𝑡))，使得𝑉(0)=0，𝑉(𝑥(𝑡))≥0且𝑉˙(𝑥(𝑡))≤𝑤(𝑢(𝑡),𝑦(𝑡))，则系统对于供给率来说是耗散的。