2.4.1 Spatio-Temporal Impulse and Frequency Responses Given a stable generatorA, a general input-output model of (2.1) is given by
φ(x, t) = Z ∞ −∞ Z Rn H(x, ζ, t−τ)d(ζ, τ)dζ dτ, (2.13)
whereH(x, ζ, t−τ), is the Green’s function (or operator kernel) of the PDE. For a system that is spatially invariant in coordinates η= [x1, . . . , xr] this can be simplified such that
y(x, t) = Z ∞ −∞ Z Rn−r Z Rr H(y, χ, η−ξ, t−τ)d(χ, ξ, τ)dξ dχ dτ, (2.14)
where y = [y1, . . . , yn−r] are the coordinates that are not spatially invariant. The spatio-
temporal impulse response is the kernel (which by abuse of notation we also refer to as
H(y, η, t)) when the input in (2.14) is an impulsive delta functionδ(y−y0, η, t). The term
spatio-temporal impulse response is used because H(y, η, t) is the solution of the PDE in (2.12) with inputδ(y−y0, η, t).
In practice, it is often easier to compute H(y, η, t) in the frequency domain. The com- bined spatial and temporal Fourier transform of an input d(η, y, t) to the system (2.1) is given by ˆ d(y, κ, ω) = Z ∞ −∞ Z G e−jωte−jκ·ηd(y, η, t)dxdt, (2.15)
whereGis the spatial group,κ∈Gandκ·η:=κTη. Since convolution in (2.14) corresponds
to multiplication in the frequency domain, it follows that
ˆ
y(y, κ, ω) = ˆH(y, κ, ω)ˆd(y, κ, ω). (2.16)
This ˆH(y, κ, ω), which corresponds to the Fourier transform ofH(y, η, t), is called the spatio- temporal frequency response of (2.1). It can easily be computed by taking the temporal Fourier transform of (2.12) and solving for ˆϕ(y, κ, ω). Then
ˆ H(y, κ, ω) = ˆCκ ³ jωI −Aˆk ´−1 ˆ Bκ. (2.17)
Equation (2.17) is also referred to as the transfer function of (2.1). Clearly one can use this spatio-temporal frequency response to compute the spatio-temporal impulse response
H(y, η, t), so both quantities contain the same information about the system’s dynamics.
However, one particular form may provide more insight into a specific system property. In the sequel, we often write the respective spatio-temporal impulse and frequency responses asH(κ, ω)(y) and H(η, t)(y) to indicate that they are in fact operator valued functions of the spatial variables y. In Section 2.4.3 we illustrate the use of both time and frequency domain responses in the computation of system gains. In general, we focus on the frequency response because it is computationally easier. In order to facilitate the gain discussion, we introduce the concept of singular values in the next section.
2.4.2 Singular Values
In this section we discuss thesingular value (Schmidt) decompositionof the spatio-temporal frequency response. The discussion is limited to stable systems because the maximal am- plification for an unstable system is always infinite and thus does not provide meaningful information. Thesingular value (Schmidt) decomposition of the spatio-temporal frequency
response is h ˆ H(κ, ω)d(κ, ω)i(y) = ∞ X m=1 σm(κ, ω)hd,pmiqm, (2.18) where {σm ≥ 0}m∈N are the singular values of H(κ, ω)(y) arranged in descending order,
and {pm}m∈N and {qm}m∈N are respectively its right and left singular functions. This
arrangement means that σ1(κ, ω) determines the worst case amplification for any input. The singular functions can be interpreted as follows. Let d := pn for some n ∈ N in (2.18). Then,
[H(κ, ω)pn(κ, ω)] (y) =σn(κ, ω)qn(y, κ, ω).
So, an input in the pn(κ, ω, y) direction produces an output in the qn(κ, ω, y) direction, whileσn(κ, ω) represents the input-output gain for a system excited in thepn direction. In light of this relationship, pn(κ, ω, y) and qn(κ, ω, y) are often referred to as the respective input and output directions. For the maximal singular value, σ1(κ, ω), the corresponding
p1(y, κ, ω) represents the most amplified input direction. The corresponding q1(y, κ, ω) is
then the output direction that has the most potential for input growth, i.e., the pair (p1,q1) correspond to the worst case input and output directions at a given (κ, ω). The notion of worst case amplification is discussed in the following system in terms of the H∞ norm.
2.4.3 System Norms and Input-Output Gain
It is not always easy to understand the dynamics of (2.1) by looking at the spatio-temporal responses derived in Section 2.4.1. In general (2.17) may be an operator valued function in both space and time. In this section we introduce the H2 and H∞ system norms [99] as a
way of quantifying behavior of (2.1) in terms of its input-output amplification or “gain”. We begin by defining the norms and then discuss how to compute the H2 norm.
Definition 2.4.1. The H∞ norm of H(y, κ, ω) is
[kHk∞](κ) := sup
ω σmax(H(κ, ω)), (2.19)
Definition 2.4.2. The H2 norm is defined as [kHk22](κ) := 1 2π Z ∞ −∞k H(κ, ω)k2HSdω= Z ∞ −∞k H(κ, t)k2HSdt, (2.20) where kHk2HS :=trace(HH∗) = ∞ X n=1 σn2(H). In the finite-dimensional (matrix) case, this is the Frobenius norm.
Definition 2.4.3. In the time domain the induced L2–to–L2 norm ofH(κ, t) is
kHkL2 := sup
kd(κ,t)k2≤1
kφ(κ, t)k2
kd(κ, t)k2
. (2.21)
In the transformed space (frequency domain) this is equivalent to theH∞ norm.
Both the H2 and H∞ norms are finite for stable systems. These norms are of interest
because they have a convenient physical interpretation. As discussed in Section 2.4.2, the
H∞norm can be interpreted as the worst case amplification of a deterministic input [46]. In
a stochastic setting, theH2norm represents the amplification of for inputsd(y, η, t) that are stochastic inyandtand harmonic inη. In the fluid mechanics literature, this is commonly referred to as the ensemble average energy density of the statistical steady-state [25].
The H2 norm of (2.1) can be computed from the solutions of the operator Lyapunov equations for the controllability and observability gramians,Xand Y [99],
AκXκ+XκA∗κ=BκBκ∗ (2.22a)
AκYκ+YκA∗κ =CκCκ∗ (2.22b) whereA∗
κ, Bκ andCκ∗ are respectively the adjoint of Aκ, Bκ and Cκ from (2.12). Then, [kHk2HS](κ) :=trace(XκCκ∗Cκ) =trace(YκBκBκ∗)