JxðÞ¼,θ∂Yðx,θÞ
∂ðx,θÞ¼∂y
∂x 1∂y
∂xn∂y
∂θ 1∂y
∂θn
⋮⋮⋮⋮
∂ynu
∂x 1∂ynu
∂xn∂ynu
∂θ 1∂ynu
∂θn0 B B B B B B @
1 C C C C C C Að 7 Þthe elements of the Jacobian matrix equals the coefficients of the
formal Taylor’s series expansion aroundt¼0 of the output sensi-
tivity derivatives with regard to initial conditions and parameters.
The sensitivity equations are:X⋆
:P
:x_ ¼ fðx,u,θÞ, xð 0 Þ¼x^0
d
dt∂x
∂x^0 i¼∂f
∂x∂x
∂x^0 i,∂x
∂x^0 iðÞ¼ (^01) n
d
dt
∂x
∂θi
¼
∂f
∂x
∂x
∂θi
þ
∂f
∂θi
,
∂x
∂θi
ðÞ¼ (^00) d
8
<
:
ð 8 Þ
The system
P⋆
can be solved iteratively generating truncated power
series solutions of desired order. Insertion into the output sensitiv-
ity expressions gives truncated power series:
d
dx^0 i
ytðÞ ¼
∂g
∂x
ðÞxtðÞ,utðÞ,θ
∂x
∂x^0 i
ðÞt
d
dθi
ytðÞ ¼
∂g
∂x
ðÞxtðÞ,utðÞ,θ
∂x
∂θi
ðÞþt
∂g
∂θi
ðÞxtðÞ,utðÞ,θ
ð 9 Þ
We may summarize the possible strategy with the following
diagram:
∂(Lifg) 0 ≤i≤ν
∂(x,θ) Symbolic
∂(Lifg) 0 ≤i≤ν
∂(x,θ) Numeric
x ̃x
θ θ ̃
u(t) u ̃(t)
x→x ̃
θ→θ ̃
u(t)→ ̃u(t)
2.3 Inverse Problem Let’s consider the computation of an approximation to a solution
of a nonlinear operator equation
FðxÞ¼y ð 10 Þ
74 Rodolfo Guzzi et al.