OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 130

of Eqs. (4.39) - (4.41) ensures the value of Eq. (4.43) for the performance
index.
Finally prove the constraint of Eq. (4.46) for the control u(t) .

To validate Eq. (4.45) it suffices to prove that

y(t) ≥˜ y 0 (t), 0 ≤ t ≤ T, (4.47)

for any control u satisfying the constraint of Eq. (4.46). From Eq. (4.32)
and the inequality u ≥− U it follows that

≥− ξ 0 (t), − ξ(t) 0 ≤ t ≤ T, (4.48)

ξ 0 (t) on the time interval 0

since ≤ t ≤ T is defined as ξ(t) for u =− U .

In a similar way, the inequality

x(t) ≥˜ x 0 (t), 0 ≤ t ≤ T, (4.49)

can be derived from Eq. (4.13), the initial conditions of Eq. (4.15) for x
and ≤ t ≤ T as the

solution of the initial-value problem of Eqs. (4.28) and (4.29) for u =− U .
Add Eqs. (4.48) and (4.49) and use Eqs. (4.33) and (4.34) for

x , the inequality u ≥− U , and the definition of x 0 for 0

y 0 and y ,

respectively, to obtain Eq. (4.47).
To derive Eq. (4.40) for the control u , proceed from Eq. (4.42), which

implies y = const and, hence, y = 0and y = 0for t > T . Use these rela-

tions together with Eq. (4.34) to obtain

ξ = x, ξ = x,

t > T. (4.50)

Then, from Eqs. (4.13) and (4.14) it follows that

ξ

c ξ + + kξ = 0 , t > T. (4.51)

This equation takes into account Assumptions 1 and 3, according to which
v(t)

= 0for t > T . Use Eqs. (4.14) and (4.51) and the definition of u as ¯

the control u for t > T to obtain

= ξ. u ¯ (4.52)

ξ

Differentiate Eq. (4.51) twice with respect to time and substitute u for ¯

into the resulting relation to arrive at Eq. (4.40).