122
M. Tirelli
...x h... |
...θh... |
...λh... |
...υ j... |
...y j... |
iiiq jiii |
t1 |
t0 | |
1 |
[D2uh] |
---- 0 |
---- -[IN] |
--- 0 |
---- 0 |
--- 0 |
----- 0 |
-- 0 |
2 |
-INH |
[W] |
0 |
0 |
...θjhIN... |
iiiiiiiiii θhT— θh 0S×J |
iiiiii T 0,iii,0 |
0 IS |
[WT] |
pï |
iλii0hiiIiJii |
iii, 0, iii | |||||
3 |
0 |
0 |
0 |
0 (5) | ||||
4 |
0 |
0 |
..iθjlN... |
T |
-[D2fj] |
0 |
0, iii, 0 |
0 |
5 |
0 |
0 |
0 |
0 |
[Dfj] |
0 |
0 |
0 |
6 |
0 |
...IJ ... |
0 |
0 |
0 |
0 |
0 |
0 |
7 |
0 |
0 |
0 |
0 |
0 |
0 |
[Σ j ∈ J2 ys ] |
HIS |
8 |
[Duh] |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
[cs ] denotes a |
diagonal matrix with typical diagonal element cs |
varying across s |
> |
Looking at Equation (2), it is immediate that, if DG has full row rank,
rDG = 0 has equations outnumbering the unknowns r = (r1 ,... ,r8) ∈
Rn+H . Thus, it is sufficient to show that the Jacobian (5) has independent
rows. An explicit formulation of (4) is:
(I) (II) |
∀h, rh1 D2uh - rh2 + rh8Duh = 0 ∀ h,j,rh Wj + rj = 0 |
NH JH |
r1 r3 |
(III) |
∀ h, — r1 + r3 WT + ∑ j^θjrj = 0 |
NH |
r2 |
(IV) |
∀j ∈ J2,rj4DfjT = 0 |
J2 |
r5 |
(V) |
∀s, ∀ j ∈ J2, — ∑ h (θ>h2s — ' ) |
NJ2 |
r4 |
— rj4Di2s fj — rj5Dsfj = 0 |
(6) | ||
(VI) |
∀ j, ∑ hrh,o (j—θh )—∑ hλ hrj=0 |
J |
r6 |
(VII) |
∑ ∑ ∑ ∑ h∩22- I г-3 h h τuj∖ ∀s ∈ S, j,h θj rh,s Ws + rh,j λs Ws |
S |
(r8) |
+ ∑j ∈ J2 (rj Wj + rlsβj) = 0 | |||
(VIII) |
∀s ∈ s, ∑h rh s + Hrl = 0 |
S |
r7 |
(IX) |
Il r II — 1 = 0 |
1 |
where each block of equations is labelled in Roman numbers on the left-hand
side, and the last two columns on the right-hand side indicate the number of
equations in each block, with the unknown variables (r ) to which they can
be matched.
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