Reduced derivative

Let X be a separable, reflexive Banach space with norm || || and fix T > 0. Let BV₋([0, T]; X) denote the space of all left-continuous functions z : [0, T] → X with bounded variation on [0, T].

For any function of time f, use subscripts +/− to denote the right/left continuous versions of f, i.e.

${\displaystyle f_{+}(t)=\lim _{s\downarrow t}f(s);}$

${\displaystyle f_{-}(t)=\lim _{s\uparrow t}f(s).}$

For any sub-interval [a, b] of [0, T], let Var(z, [a, b]) denote the variation of z over [a, b], i.e., the supremum

${\displaystyle \mathrm {Var} (z,[a,b])=\sup \left\{\left.\sum _{i=1}^{k}\|z(t_{i})-z(t_{i-1})\|\right|a=t_{0}<t_{1}<\cdots <t_{k}=b,k\in \mathbb {N} \right\}.}$

The first step in the construction of the reduced derivative is the “stretch” time so that z can be linearly interpolated at its jump points. To this end, define

${\displaystyle {\hat {\tau }}\colon [0,T]\to [0,+\infty );}$

${\displaystyle {\hat {\tau }}(t)=t+\int _{[0,t]}\|\mathrm {d} z\|=t+\mathrm {Var} (z,[0,t]).}$

The “stretched time” function τ̂ is left-continuous (i.e. τ̂ = τ̂₋); moreover, τ̂₋ and τ̂₊ are strictly increasing and agree except at the (at most countable) jump points of z. Setting T̂ = τ̂(T), this “stretch” can be inverted by

${\displaystyle {\hat {t}}\colon [0,{\hat {T}}]\to [0,T];}$

${\displaystyle {\hat {t}}(\tau )=\max\{t\in [0,T]|{\hat {\tau }}(t)\leq \tau \}.}$

Using this, the stretched version of z is defined by

${\displaystyle {\hat {z}}\in C^{0}([0,{\hat {T}}];X);}$

${\displaystyle {\hat {z}}(\tau )=(1-\theta )z_{-}(t)+\theta z_{+}(t)}$

where θ ∈ [0, 1] and

${\displaystyle \tau =(1-\theta ){\hat {\tau }}_{-}(t)+\theta {\hat {\tau }}_{+}(t).}$

The effect of this definition is to create a new function ẑ which “stretches out” the jumps of z by linear interpolation. A quick calculation shows that ẑ is not just continuous, but also lies in a Sobolev space:

${\displaystyle {\hat {z}}\in W^{1,\infty }([0,{\hat {T}}];X);}$

${\displaystyle \left\|{\frac {\mathrm {d} {\hat {z}}}{\mathrm {d} \tau }}\right\|_{L^{\infty }([0,{\hat {T}}];X)}\leq 1.}$

The derivative of ẑ(τ) with respect to τ is defined almost everywhere with respect to Lebesgue measure. The reduced derivative of z is the pull-back of this derivative by the stretching function τ̂ : [0, T] → [0, T̂]. In other words,

${\displaystyle \mathrm {rd} (z)\colon [0,T]\to \{x\in X|\|x\|\leq 1\};}$

${\displaystyle \mathrm {rd} (z)(t)={\frac {\mathrm {d} {\hat {z}}}{\mathrm {d} \tau }}\left({\frac {{\hat {\tau }}_{-}(t)+{\hat {\tau }}_{+}(t)}{2}}\right).}$

Associated with this pull-back of the derivative is the pull-back of Lebesgue measure on [0, T̂], which defines the differential measure μ_z:

${\displaystyle \mu _{z}([t_{1},t_{2}))=\lambda ([{\hat {\tau }}(t_{1}),{\hat {\tau }}(t_{2}))={\hat {\tau }}(t_{2})-{\hat {\tau }}(t_{1})=t_{2}-t_{1}+\int _{[t_{1},t_{2}]}\|\mathrm {d} z\|.}$

• The reduced derivative rd(z) is defined only μ_z-almost everywhere on [0, T].
• If t is a jump point of z, then

${\displaystyle \mu _{z}(\{t\})=\|z_{+}(t)-z_{-}(t)\|{\mbox{ and }}\mathrm {rd} (z)(t)={\frac {z_{+}(t)-z_{-}(t)}{\|z_{+}(t)-z_{-}(t)\|}}.}$

• If z is differentiable on (t₁, t₂), then

${\displaystyle \mu _{z}((t_{1},t_{2}))=\int _{t_{1}}^{t_{2}}1+\|{\dot {z}}(t)\|\,\mathrm {d} t}$

and, for t ∈ (t₁, t₂),

${\displaystyle \mathrm {rd} (z)(t)={\frac {{\dot {z}}(t)}{1+\|{\dot {z}}(t)\|}}}$,

• For 0 ≤ s < t ≤ T,

${\displaystyle \int _{[s,t)}\mathrm {rd} (z)(r)\,\mathrm {d} \mu _{z}(r)=\int _{[s,t)}\mathrm {d} z=z(t)-z(s).}$

• Mielke, Alexander; Theil, Florian (2004). "On rate-independent hysteresis models". NoDEA Nonlinear Differential Equations Appl. 11 (2): 151–189. ISSN 1021-9722. MR2210284