= Storage function
{title2=$S(x)$}
A <storage function> assigns a nonnegative quantity to the state of a <dynamical system> so that its increase is bounded by the integrated <supply rate>. In <dissipativity (control theory)>, it must satisfy
$$
S(x(t_1))-S(x(t_0))\leq\int_{t_0}^{t_1}w(u(t),y(t))\,dt
$$
for every allowed history and interval. A <storage function> may represent physical <energy>, but it can also certify an abstract input–output bound. <Jan C. Willems>'s theory relates possible storage functions to recoverable supply, rather than selecting an arbitrary positive expression.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[General Theory, storage functions and available storage]{ref}
For a differentiable <storage function> and state equation $\dot x=f(x,u)$, the pointwise inequality
$$
\nabla S(x)^Tf(x,u)\leq w(u,y)
$$
is sufficient: integrating it along a solution proves the interval statement. Positivity alone is insufficient; the derivative must be checked against the specified <supply rate>.
An original dimensionless example makes this check exact. For $\dot x=-x+u$, output $y=x$, and <supply rate> $w=uy=ux$, try $S_a(x)=a x^2/2$ with $a>0$. Its derivative is
$$
\dot S_a=-a x^2+a xu.
$$
The difference from supply is $\dot S_a-ux=-a x^2+(a-1)xu$. Requiring it to be nonpositive for every real $u$ and fixed nonzero $x$ forces $a=1$. That value works because $\dot S_1=ux-x^2\leq ux$. Thus even within this simple positive quadratic family, the supply law fixes the admissible scale.
A <storage function> also limits extraction. Rearranging the interval inequality gives
$$
-\int_{t_0}^{t_1}w\,dt\leq S(x(t_0))-S(x(t_1))\leq S(x(t_0)).
$$
Starting in a given state, one cannot withdraw more net supply than the initial storage certificate permits. A chosen <storage function> need not make that upper bound attainable.
When unforced operation makes the <supply rate> zero, storage cannot increase. This resembles a <Lyapunov function>, but nonincrease alone does not prove <asymptotic stability>: the function may fail to distinguish some states, or its derivative may vanish along persistent motion. The storage balance and a convergence proof are separate claims.
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