Storage function ($S(x)$)

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, it must satisfyfor 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.^[ref]

For a differentiable storage function and state equation $\dot{x}=f(x,u)$, the pointwise inequalityis 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 isThe 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 ${\dot{S}}_1=ux-x^2\le ux$. 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 givesStarting 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.