The information of differential equations (18) is attached onto the minimization of the objective functional by
adjoint variable [lambda].
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 [28].
Adjoint variable 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 functions.