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.
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