Security for Multihop Wireless Networks

Securing the Control of Euler–Lagrange Systems  ◾  363



xkxs

11

=− +

(13.16)

where ||s|| <

δ

for t > k

2

τ. Note that Equation 13.16 can be regarded as a linear

system with x

1

as state and s as input. Clearly, the system (13.16) is bound-input-

bounded-output since its system matrix has all eigenvalues strictly on the left-half-

plane. erefore, x

1

is bounded ultimately as the input s is bounded. To analyze

the bound on x

1

in quantity, we construct the measurement

Vxx

T

=

11

.

e time

derivative of V is





Vxxkxx xs

kx xxsk

Vx

s

TTT

TT

==−+

=− +≤−+

222

2222

11 11 1

11

11

|| || || ||

==− +≤−+

∀>

22 22

2

kV Vs kV V

tk

|| || δ

τ

(13.17)

Defining

WV

= , the differential inequalit