ABSTRACT
In the generic abstract parameter estimation problem, we consider a dynamic model of the form (9.1) where the operators A and D and possibly the input f depend on some unknown (i.e., to be estimated) functional parameters q in an admissible family Q ⊂ C1(0, T ;L∞(Ω;Q)) of parameters. Here Ω is the underlying set on which the functions of H,V, VD are defined (e.g., the spatial set Ω = (0, l) in the heat, transport, and beam examples). We assume that the time dependence of the operators A and D are through the time dependence of the parameters q(t) ∈ L∞(Ω;Q) where Q ⊂ Rp is a given constraint set for the values of the parameters. That is, A(t) = A1(q(t)), D(t) = A2(q(t)) so that we have
y¨(t) +A2(q(t))y˙(t) +A1(q(t))y(t) = f(t, q) in V ∗
y(0) = y0, y˙(0) = y1. (10.1)
Thus we introduce the sesquilinear forms σ1, σ2 by
a(t;φ, ψ) ≡ σ1(q(t))(φ, ψ) = 〈A1(q(t))φ, ψ〉V ∗,V , (10.2) d(t;φ, ψ) ≡ σ2(q(t))(φ, ψ) = 〈A2(q(t))φ, ψ〉V ∗D,VD . (10.3)
It will be convenient in subsequent arguments to use the notation
σ˙i(q)(φ, ψ) ≡ d dt σi(q)(φ, ψ) (10.4)
which in the event that σi is linear in q becomes σ˙i(q)(φ, ψ) = σi(q˙)(φ, ψ). For example, in the Euler-Bernoulli example of (9.18), we would have
σ1(q(t))(φ, ψ) =
∫ l 0 E˜I(t, ξ)φ′′(ξ)ψ′′(ξ)dξ (10.5)
σ2(q(t))(φ, ψ) =
∫ l 0 C˜DI(t, ξ)φ
′′(ξ)ψ′′(ξ)dξ, (10.6)
where q = (E˜I, C˜DI) ∈ C1(0, T ;L∞(0, l;R2+)). Note that in this case we do have σ˙i(q)(φ, ψ) = σi(q˙)(φ, ψ).
