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→Dissipative systems in control theory
== 控制论中的耗散系统 Dissipative systems in control theory ==
== 控制论中的耗散系统 Dissipative systems in control theory ==
[[Jan Camiel Willems|Willems]] first introduced the concept of dissipativity in systems theory<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> to describe 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
[[Jan Camiel Willems|Willems]] first introduced the concept of dissipativity in systems theory<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> to describe 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