]> 2.8 Solution of the Lyapunov equation

2.8 Solution of the Lyapunov equation

To find the kernel h of the integral operator (2.43), we put x(θ) = δ(θ a), v(θ) = δ(θ b), with 0 a,b 1, where δ is the Dirac pseudofunction, in (2.45), which yields
h(a,b) = h(b,a) = n= Res λ=λn sinh λa λ(cosh λ K) sinh λb λ(cosh λ K) = = n= 2 sin λna sinh λnb λn sinh λn(cos λn K),0 a,b 1 ,

where λn = μn (n ) are given by (2.35). In particular,

h(a,b) = 2 1 K2 n= sin[a(φ 2nπ)] sinh[b(φ 2nπ)] (φ 2nπ)[cosh(φ 2nπ) cos(φ 2nπ)] ,

where K < 1 and φ = arccos K. Employing the partial–fraction expansion of the meromorphic function z(sin az sinh bz)(z[cosh z cos z]) [56, Problem 5.2.1], we obtain

h(a,b) = 2 1 K2 n= ab φ 2nπ + k=1 n= Bk φ 2nπ zk+ + k=1 n= Bk¯ φ 2nπ + zk¯ + k=1 n= Bk φ 2nπ + zk+ + k=1 n= Bk¯ φ 2nπ zk¯ (2.54)

where Bk = (sin azk sinh bzk)(sinh zk + sin zk) and zk = (1 + j)kπ (k ). Taking into account the identity [56, Problem 5.2.4]

n= 1 φ 2nπ = 1 2 cot φ 2

and eliminating φ in (2.54), and after some transformations, we obtain the final formula for h, valid for all K ( cosh π, 1):

h(a,b) = ab 1 K + + 1 π k=1cos(a b)kπ cosh(a + b)kπ cosh(a b)kπ cos(a + b)kπ k sinh kπ[cosh kπ (1)kK] = = ab 1 K + 2 π k=1 [cos(a b)kπ][e(a+b2)kπ + e(a+b+2)kπ] k(1 e2kπ)[1 + e2kπ 2K(1)kekπ] [cos(a + b)kπ][e(ab2)kπ + e(ab+2)kπ] k(1 e2kπ)[1 + e2kπ 2K(1)kekπ]0 a,b 1 (2.55)

The series in (2.55) converges in square a + b < 2. From the divergence of the harmonic series and the estimate

sinh kπ cosh kπ (1)kK sinh π cosh π + K,k ,

it follows that the series in (2.55) is divergent at a = b = 1. Hence, the kernel h is unbounded: h(1, 1) = . The plots of the kernel h for two admissible values of K are depicted in Figures 2.5 and 2.6.


PIC

Figure 2.5: Plot of the kernel h = h(a,b) for K = 0.75


PIC

Figure 2.6: Plot of the kernel h = h(a,b) for K = 10

Numerical calculations show that the folding of h increases for K decreasing from 0 to cosh π. For K < 1 the kernel h may take negative values.

Remark 2.8.1. Using (2.25) and (2.43), it is possible to characterize the kernel h by the integral equation

01[k(τ,ϑ)h(ϑ,θ) + h(ϑ,τ)k(τ,θ)]dτ = d(θ)d(ϑ),0 θ,ϑ 1 ,

here, k denotes the kernel of 𝒜1 (see (2.31)), and d is defined by (2.8). An approximate solution of this equation can be found with the aid of the inflation technique [46] which allows us to pass to the matrix Lyapunov equation.