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第93行：
第93行： − + − Willems首先在系统理论[8]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）、其输入𝑢（𝑡）和其输出𝑡（𝑡）所描述的动力系统，给出了输入输出关系式。如果存在一个连续可微的存储函数𝑉（𝑥（𝑡）），使得𝑉（𝑥（𝑡））≥0且𝑉（𝑥（𝑡））≤𝑢（𝑡），𝑦（𝑡））。+ 第124行：
第124行： − 这个概念与Lyapunov稳定性有很强的联系，其中存储函数可以在一定的能控性和可观测性条件下发挥Lyapunov函数的作用。+ 第134行：
第134行： − − − ------------ − − In systems theory the concept of dissipativity was first introduced by [[Jan Camiel Willems|Willems]],<ref>{{cite journal |last1=Willems |first1=J.C. |title=Dissipative dynamical systems part 1: General theory |journal=Arch. Rational Mech. Anal. |date=1972 |volume=45 |issue=5 |page=321 |doi=10.1007/BF00276493 |bibcode=1972ArRMA..45..321W |hdl=10338.dmlcz/135639 |url=http://dml.cz/bitstream/handle/10338.dmlcz/135639/Kybernetika_41-2005-1_5.pdf }}</ref> which describes dynamical systems by input-output properties. Considering a dynamical system described by its state <math> x(t) </math>, its input <math>u(t)</math> and its output <math>y(t)</math>, the input-output correlation is given a supply rate <math> w(u(t),y(t))</math>. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function <math> V(x(t))</math> such that <math>V(0)=0</math>, <math>V(x(t))\ge 0 </math> and − − In systems theory the concept of dissipativity was first introduced by Willems, which describes dynamical systems by input-output properties. Considering a dynamical system described by its state <math> x(t) </math>, its input <math>u(t)</math> and its output <math>y(t)</math>, the input-output correlation is given a supply rate <math> w(u(t),y(t))</math>. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function <math> V(x(t))</math> such that <math>V(0)=0</math>, <math>V(x(t))\ge 0 </math> and − − :<math> \dot{V}(x(t)) \le w(u(t),y(t))</math>.<ref>{{cite book |last1=Arcak |first1=Murat |last2=Meissen |first2=Chris |last3=Packard |first3=Andrew |title=Networks of Dissipative Systems |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-29928-0 }}</ref> − − 在系统论中，耗散性的概念是由 Jan Camiel Willems 首先提出的，它通过输入输出的性质来描述动态系统。考虑一个由状态 x (t) / math 描述的动力系统，它的输入数学 u (t) / math 和输出数学 y (t) / math，输入输出相关性被给出一个供给率数学 w (u (t) ，y (t) / math。如果存在一个连续可微的存储函数数学 v (x (t)) / 这样的数学 v (0)0 / math，数学 v (x (t)) ge0 / math，那么系统相对于供给率是耗散的 − − − − − <math> \dot{V}(x(t)) \le w(u(t),y(t))</math>. − − Math dot { v }(x (t)) le w (u (t) ，y (t)) / math. − − − − − − As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate <math> w(u(t),y(t)) = u(t)^Ty(t) </math>. − − As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate <math> w(u(t),y(t)) = u(t)^Ty(t) </math>. − − 作为耗散性的一个特例，如果上述耗散性不等式对于被动供给率数学 w (u (t) ，y (t)) u (t) ^ Ty (t) / math 成立，则系统被称为被动系统。 − − − − − − The physical interpretation is that <math>V(x)</math> is the energy stored in the system, whereas <math>w(u(t),y(t))</math> is the energy that is supplied to the system. − − The physical interpretation is that <math>V(x)</math> is the energy stored in the system, whereas <math>w(u(t),y(t))</math> is the energy that is supplied to the system. − − 物理解释是 math v (x) / math 是存储在系统中的能量，而 math w (u (t) ，y (t)) / math 是提供给系统的能量。 − − − − − − This notion has a strong connection with [[Lyapunov stability]], where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions. − − This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions. − − 这个概念与李雅普诺夫稳定性有很强的联系，在动力系统的可控性和可观测性的特定条件下，存储函数可以扮演 Lyapunov 函数的角色。 − − − − − − Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by [[Vasile M. Popov|V.M. Popov]], [[Jan Camiel Willems|J.C. Willems]], D.J. Hill, and P. Moylan. In the case of linear invariant systems{{clarify|reason=Is this the same as a "linear time-invariant system" as in the Wikipedia articles "LTI system theory"?|date=April 2015}}, this is known as positive real transfer functions, and a fundamental tool is the so-called [[Kalman–Yakubovich–Popov lemma]] which relates the state space and the frequency domain properties of positive real systems{{clarify|reason=What is a positive real system?|date=April 2015}}.<ref>{{cite book|url=https://www.springer.com/978-1-84628-892-0|title=Process Control - The Passive Systems Approach| last1=Bao| first1=Jie| last2=Lee| first2=Peter L.| authorlink2=Peter Lee (engineer)| publisher=[[Springer Business+Science Media|Springer-Verlag London]]|year=2007|doi=10.1007/978-1-84628-893-7|isbn=978-1-84628-892-0}}</ref> Dissipative systems are still an active field of research in systems and control, due to their important applications. − − Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by V.M. Popov, J.C. Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems. Dissipative systems are still an active field of research in systems and control, due to their important applications. − − 耗散性理论对于线性和非线性系统的反馈控制律设计具有一定的参考价值。耗散系统理论已由 v.m。波波夫，J.C.Willems，d.j。希尔和 p. 莫伊兰。在线性不变系统的情况下，这被称为正实传递函数，一个基本的工具是所谓的 Kalman-Yakubovich-Popov 引理，它把正实系统的状态空间和频域性质联系起来。由于耗散系统的重要应用，它仍然是系统与控制研究的一个活跃领域。
→控制论中的耗散系统 Dissipative systems in control theory
Willems first introduced the concept of dissipativity in systems theory[8] 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。一个由其状态𝑥（𝑡）、其输入𝑢（𝑡）和其输出𝑡（𝑡）所描述的动力系统，给出了输入输出关系式。如果存在一个连续可微的存储函数𝑉（𝑥（𝑡）），使得𝑉（𝑥（𝑡））≥0且𝑉（𝑥（𝑡））≤𝑢（𝑡），𝑦（𝑡））。
This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.
This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.
这个概念与Lyapunov稳定性有很强的联系，其中存储函数可以在一定的能控性和可观测性条件下发挥李雅普诺夫函数的作用。
粗略地说，耗散性理论对于线性系统的设计是有用的。耗散系统理论已经由V.M.Popov、J.C.Willems、D.J.Hill和P.Moylan讨论过。在线性不变系统的情况下，这被称为正实传递函数，一个基本的工具就是所谓的Kalman-Yakubovich-Popov引理，它联系了正实系统的状态空间和频域特性。
粗略地说，耗散性理论对于线性系统的设计是有用的。耗散系统理论已经由V.M.Popov、J.C.Willems、D.J.Hill和P.Moylan讨论过。在线性不变系统的情况下，这被称为正实传递函数，一个基本的工具就是所谓的Kalman-Yakubovich-Popov引理，它联系了正实系统的状态空间和频域特性。
== Quantum dissipative systems ==
== Quantum dissipative systems ==