The information of differential equations (18) is attached onto the minimization of the objective functional by adjoint variable
We denote the adjoint variable
by [p.sup.(3)] in the third interval, by [p.sup.(2)] in the second interval, and by [p.sup.(3)] in the first interval.
An additional argument holds the adjoint variable
of the result of the function.
If [lambda](t) solved for the time series group (i.e., the transient Adjoint variable
), the design variable sensitivity ([[alpha].sub.k] differentiation) forms the remaining terms of Equations (6) and (7) and can be calculated by the following arithmetic computation (time integration).
And we define the state and adjoint variable
S(t), I(t), T(t), R(t), [[lambda].sub.1], A,f,f and the control u and v in terms of nodal points [S.sub.i], [I.sub.i], [T.sub.i], [R.sub.i], [[lambda].sup.i.sub.1], [[lambda].sup.i.sub.2], [[lambda].sup.i.sub.3], [[lambda].sup.i.sub.4], [u.sup.i] and [v.sup.i].
Then there exists a piecewise differentiable adjoint variable
In this paper, computational methods for sensitivity matrix are discussed and a novel sensitivity based damage detection method in time-domain referred to as "Adjoint variable
method," is developed.
The Adjoint variable
method (AVM) has been proposed for the acceleration of derivative-based optimization of microwave devices .
method for the thermal design of eutectic directional solidification processes in an open-boat configuration, -Numer.
Assuming sufficient smoothness, the design sensitivity of F with respect to the design variable [[phi].sub.i] may be calculated via the adjoint variable
method (see, e.g., Ref.
Sincethestatevariableisstochastic, thereisapair of processes in the adjoint variable
, one of which multiplies the diffusion of the state and another the drift.
where L(C, I, [u.sub.1], [u.sub.2], [u.sub.3], t) = [b.sub.1]C + [b.sub.2]I + (1/2) [[summation].sup.3.sub.=1] [w.sub.i][u.sup.2.sub.i], [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3], [[lambda].sub.4], and [[lambda].sub.5] are the adjoint variable