= Mechanical available storage
<Mechanical available storage> is the supremum of net <work> that can be extracted from a system's current state through its allowed mechanical terminals. If $Fv$ is entering <power> and the initial state is $x_0$, define
$$
S_a(x_0)=\sup_{T\ge0,\;\text{admissible motions}}
\left[-\int_0^T F(t)v(t)\,dt\right].
$$
The zero-duration choice gives $S_a\ge0$. Any nonnegative <mechanical energy storage function> satisfying the <mechanical dissipation inequality> bounds the extracted work, so $S_a(x_0)\le E(x_0)$. Available storage is thus an operational recoverability quantity; it need not equal every form of internal stored energy. The general supply/storage construction was developed by <Jan C. Willems>.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[Abstract, recoverable-storage interpretation; time-varying extension, Definition1′ and subsequent available-storage formula]{ref}
Consider a <spring> and <dashpot> in parallel, with $F=kx+c\dot x$ and initially $x=x_0$. Unload to $x=0$ at constant speed over duration $T$, so $\dot x=-x_0/T$. The outgoing <work> is
$$
W_{\rm out}=\frac12kx_0^2-\frac{cx_0^2}{T}.
$$
The first term is released <elastic potential energy>; the second is lost in the <dashpot>. For $k=1000\,\mathrm{N/m}$, $c=10\,\mathrm{N\,s/m}$ and $x_0=0.10\,\mathrm m$, the initial spring energy is $5\,\mathrm J$. A one-second unloading recovers $4.90\,\mathrm J$ and a ten-second unloading recovers $4.99\,\mathrm J$. With unrestricted duration and suitable terminal control, these paths approach the energy upper bound, proving $S_a=5\,\mathrm J$ for this ideal example. A supremum need not be attained in finite time.
The allowed operations are essential. If the same example must finish at zero extension within one second, the <Cauchy-Schwarz inequality> gives $\int_0^T\dot x^2dt\ge x_0^2/T$, so the best recoverable work under that extra requirement is $4.90\,\mathrm J$. A physically inaccessible internal motion may store energy without allowing its recovery at the chosen terminals. Thus stating the initial state, usable ports, time and actuator limits is part of an available-storage claim, not an optional qualification.
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