Page 101 - Control Theory in Biomedical Engineering

88 Control theory in biomedical engineering

Lemma 1. (Gronwall’s lemma; Halanay, 1966, p. 9) Let x, ψ, and χ be real
continuous functions defined in [a, b], χ 0 for t [a, b]. We suppose that on
R t
[a, b] we have the inequality xðtÞ ψðtÞ + χðsÞxðsÞds: Then
a

R t
R T χðξÞdξ
xðtÞ ψðtÞ + χðsÞψðsÞe s ds in [a, b].
a
Therefore, we arrive at the following proposition.
Proposition 1. Let (E(t), T(t)) be a solution of system (2), then E(t) < M 1
and T(t) < M 2 for all sufficiently large time t, where
σ Z σ Z t
t
M 1 ¼ Eð0Þ + expðδtÞ + ρe δðτ + sÞ Eð0Þ + e δs exp ρe δðτ + ξÞ dξ ds,
δ 0 δ s

1
M 2 ¼ max ,Tð0Þ :
β
Proof. Let (E, T) denotes the solution of model (2). From the second
equation of system (2), we have dT r 2 TðtÞð1 βTðtÞÞ. Thus, T(t) may
dt
be compared with the solution of

dX
¼ r 2 XðtÞð1 βXðtÞÞ, with Xð0Þ¼ Tð0Þ
dt
This proves that T(t) < M 2 . From the first equation of system (2), we obtain

t ρEðs τÞTðs τÞ
Z
EðtÞ¼ expð δtÞ Eð0Þ + σ + μEðs τÞTðs τÞ expðδsÞds :
0 η + Tðs τÞ

To show that E(t) is bounded, we use the generalized Gronwall lemma.
Since T < 1 and expð δtÞ ð0,1Š, we have
η + T
σ Z t
EðtÞ E 0 + expðδtÞ + ρEðs τÞexpðδsÞds:
δ 0

The generalized Gronwall lemma gives E(t) < M 1 where M 1 is uniformly
bounded. It follows that if (E, T) is a solution of Eq. (2), then (E, T) <
(M 1 , M 2 ) for all t. This shows that the solutions of model (2) are uniformly
bounded. This completes the proof. ▪
R
t
From Eq. (1a) and the solution TðtÞ¼ Tð0Þexp ½ r 2 ð1 βTðsÞÞ
0
EðsފdsÞ, we arrive at the following result: