= Dissipativity
{disambiguate=control theory}
<Dissipativity (control theory)> expresses an input–output balance for a <dynamical system>: the increase of a nonnegative <storage function> cannot exceed the accumulated <supply rate>. For state $x$, <control input> $u$, <plant output> $y$ and chosen <supply rate> $w(u,y)$, the requirement is
$$
S(x(t_1))-S(x(t_0))\leq\int_{t_0}^{t_1}w(u(t),y(t))\,dt
$$
for every admissible trajectory and time interval, with $S\geq0$. The <storage function> and <supply rate> need not be literal physical <energy> and <power>; choosing them makes a particular performance statement precise. <Jan C. Willems>'s1972paper develops this general storage–supply formulation.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[General Theory, storage functions and dissipation inequality]{ref}
An original mechanical example is a mass with viscous damping, driven by an applied <force> $u$:
$$
m\dot v=u-bv,\qquad y=v,\qquad m>0,\quad b\geq0.
$$
Choose the <kinetic energy> $S(v)=mv^2/2$ and entering <power> $w(u,y)=uy$. Then
$$
\dot S=mv\dot v=uv-bv^2\leq uv.
$$
Integration gives the required inequality. The nonnegative integral $\int bv^2\,dt$ accounts for the mechanical <energy> dissipated by the damper. With no applied <force>, the stored <kinetic energy> cannot increase. A <dashpot> driven at constant <velocity> provides an experimental interpretation: supplied <power> is dissipated even when the mass's stored <kinetic energy> remains constant.
A different choice tests an input–output gain. If a <storage function> establishes the inequality for
$$
w(u,y)=\gamma^2\|u\|^2-\|y\|^2,
$$
then $S(x(0))=0$ and nonnegative final storage imply
$$
\int_0^T\|y\|^2\,dt\leq\gamma^2\int_0^T\|u\|^2\,dt.
$$
The physical power balance and this squared-signal bound use the same logical structure. Neither inequality, by itself, proves that every internal state converges to zero: a hidden lossless motion can retain storage.
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