Applying these formulae we find for ra = 0

[a(x) — a(x)]Z(x, x') dx + o(||a — a||), [a(x) — x(x)]r (x')Z*(x, x') dx dx',

Z*(x, x') = —g*(x, x')ga(x), are referred to as densities of sensitivity. They determine uniquely the functional derivatives of y with respect to a.

Estimations of feedback influence. In climate models it is common practice to accept simplifications based on adopting empirical values of the parameter a depending on a system state. The use of these models for prediction entails inevitable errors due to variations of a as y changes.

An allowance for a dependence of a on y in such cases is described as an allowance for a feedback influence. One possible method of obtaining a priori estimates of the feedback influence is the application of the above formulae (6.1.9) and (6.1.9'). A feedback can be taken into account in Equation (6.1.2) if the latter is supplemented by a relation describing the dependence of a on y. Then, instead of Equation (6.1.2), we have the system

where / is a prescribed operator and X is a new parameter describing the intensity of the feedback.

The state y of the system without an allowance for feedback is derived from solution of the equation

The problem is to estimate the quantities y — y and R(y, a) — R(y, a), where R is a non-linear functional satisfying the conditions (6.1.6) and (6.1.8). In this case the formulae (6.1.9') and (6.1.3) cannot be used directly because the parameter variations depend on system state variations. But use could be made of the old formulae representing X as a new parameter. Then, instead of the relations (6.1.3) and (6.1.9') we have y(a) - y(6c) = (X + o(\X\),

8y J da

VR = X\^*, — f(y)) + X(ra,f(y)), where, as before, the function C* is determined by Equation (6.1.5'), and derivatives dG/dy and dG/da are evaluated at the point y, a.

Following Cacuci and Hall (1984) we consider the problem of determination of temperature T complying with the conditions

^ = -ar4 + P, T\t=a = Ta, a < t < b. (6.1.14)

As a measure of sensitivity we use the functional R(T, a) = aTA df assuming that the feedback is prescribed by the relation a = a + \{T — Ta)4 (6.1.15)

and find the estimates of the quantity R(T, a) — R(f, a), where T is a solution to the problem (6.1.14) and (6.1.15) for a = a.

Let us denote © = T - Ta, © = f - Ta, G(0, a) = d©/dt + a(® + Ta)4 - p, and take as a domain of definition of the operator G a set of pairs (©, X) where the function 0 satisfies the conditions

X is a numerical parameter from the interval [ — 1,1]. Then, the functional R is given by the relation

R(&, X) = (a + /©)(© + Tay df, the difference A = R(T, a) — R{T, a) to be estimated is represented in the form

rx df, r& = 4a(@ + Ta)3, rx = &{@+Ta)\ and Equation (6.1.5') is reduced to the following problem relative to £*:

A solution to this problem has the form

By virtue of (6.1.13) we obtain

Sensitivity of solutions to time-dependent problems. Let us consider the sensitivity of a solution to the Cauchy problem as applied to Equation (6.1.1). In this case the direct equations of sensitivity coincide with the equations for variations. For simplicity we write down these equations assuming that initial data are independent of the parameter a.

Let the solution y(t) being investigated correspond to initial data y(0) and the value a of the parameter a. Let

where v'x is the resolving operator of Equation (6.1.1). Formal differentiation of Equation (6.1.1) yields df

where IY is the identical operator in Y; the operators A(t) and B(t) are defined as

Equations for higher derivatives with respect to the parameter a and to initial data are found in an analogous fashion. Each of these derivatives is a solution to the problem

with the functions /i(0) and / defined in a proper way. It should be noted that the equations appearing in (6.1.17), (6.1.18) and (6.1.10) are linear but non-autonomous in the case where the field v in (6.1.1) is independent of time explicitly.

The problem under consideration is equivalent to the functional equation

dy d t u(y, a), where the desired function y has to satisfy the condition

Let R(y, a) be a functional of a solution to the Cauchy problem complying with the conditions

lly-^lrdi

In this case the basic relations of the theory of sensitivity are written as follows:

y(t, a) - y(t, a) = Ç(a - a) + o(||a - â||), R(y, a) - R(y, a) = VR + o(\\a - a||), 'r/ Ôv

where the function (* is a solution to the problem d C* 1

The relations (6.1.22) and (6.1.23) are fulfilled a fortiori for smoothly bounded fields v. If a field v and an operator A(t) of linearization are unbounded (as is the case for partial differential equations), the problems (6.1.18) and (6.1.23) are ill-posed in general, and the formulae (6.1.20)-(6.1.22) are invalid. A sufficient condition of correctness for the problems (6.1.18) and (6.1.23) is, for example, the Hille-Iosida condition

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