Manuscript Em06-B@1 · Em06-B-v1.tex · theorem · 61e38859

Theorem: Explicit input-stability allowance

Quantitative stability of the classical inputs › Metric, switch, and observation-test changes · lines 1178–1224 · proof included or linked

Em06-B@1#thm:input-stability, Em06-B@1#eq:input-difference, Em06-B@1#eq:bounded-input-difference, Em06-B@1#eq:switch-difference

Back to manuscript

With the same fixed constant type as Theorem thm:estimate,

∥ v_{1} - v_{0} ∥ \leq C [p_{6} (f_{1} - f_{0}) H_{6} (h_{1}) (1 + c_{3} (χ_{1})) + p_{6} (f_{0}) H_{6} (h_{1} - h_{0}) (1 + c_{3} (χ_{1})) + p_{6} (f_{0}) H_{6} (h_{0}) c_{3} (χ_{1} - χ_{0})] .

In particular, on a family bounded by $F, H, X$ ,

∥ v_{1} - v_{0} ∥ \leq C [(1 + X) (H p_{6} (f_{1} - f_{0}) + F H_{6} (h_{1} - h_{0})) + F H c_{3} (χ_{1} - χ_{0})] .

The switch difference is exactly

q_{f, h, χ_{1}} - q_{f, h, χ_{0}} = - Q_{(χ_{1} - χ_{0}) h} u_{f} .

The coordinate prescription does not vary with the switch.

Proof. The map is linear in $f$ , linear in $h$ for fixed $χ$ , and its coordinate term is independent of $χ$ . Subtract first the $f$ inputs, then the $h$ inputs, and finally the switches. Apply (eq:main-estimate) to the first two differences.

For the third, (eq:switch-difference) is exact. The same product-rule estimate used in Appendix app:PDE gives

∥ k_{Q_{(χ_{1} - χ_{0}) h_{0}} u_{f_{0}}} ∥ \leq C p_{6} (f_{0}) c_{3} (χ_{1} - χ_{0}) H_{3} (h_{0}) .

Replace $H_{3}$ by $H_{6}$ . This proves both allowances.

Displayed formulas in this passage

Em06-B@1#eq:input-difference · line 1182
$∥ v_{1} - v_{0} ∥ \leq C [p_{6} (f_{1} - f_{0}) H_{6} (h_{1}) (1 + c_{3} (χ_{1})) + p_{6} (f_{0}) H_{6} (h_{1} - h_{0}) (1 + c_{3} (χ_{1})) + p_{6} (f_{0}) H_{6} (h_{0}) c_{3} (χ_{1} - χ_{0})] .$
Em06-B@1#eq:bounded-input-difference · line 1194
$∥ v_{1} - v_{0} ∥ \leq C [(1 + X) (H p_{6} (f_{1} - f_{0}) + F H_{6} (h_{1} - h_{0})) + F H c_{3} (χ_{1} - χ_{0})] .$
Em06-B@1#eq:switch-difference · line 1202
$q_{f, h, χ_{1}} - q_{f, h, χ_{0}} = - Q_{(χ_{1} - χ_{0}) h} u_{f} .$
unlabeled · line 1219
$∥ k_{Q_{(χ_{1} - χ_{0}) h_{0}} u_{f_{0}}} ∥ \leq C p_{6} (f_{0}) c_{3} (χ_{1} - χ_{0}) H_{3} (h_{0}) .$

Theorem: Explicit input-stability allowance

Displayed formulas in this passage

Nearby in this section