OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 129

subject to the initial conditions

k ξ 0 (T ),

¯

ck ξ 0 (T ),

¯ =− =

u(T ) u(T ) (4.41)

provides an optimal control for Problem 4.2. The behavior of the coordinate
y under this control is given by

y 0 (t) ˜ for 0 ≤ t ≤ T,

y 0 (t) = (4.42)

for t > T, y 0 (T ) ˜

and the minimum value of the performance index is determined as

J 1 (u 0 ) =˜ y 0 (T ∗ ). (4.43)

This proposition suggests an algorithm for solving Problem 4.2. The
algorithm involves the following steps.

Step 1. Use an appropriate technique from Chapter 3 to solve Prob-
lem 4.4.
Step 2. Check the relations of Eqs. (4.36) and (4.37).
Step 3. Use the relations of Eqs. (4.33) and (4.35) to form the function

1

exp

c t ,

U
k

k

y 0 (t) ˜ =˜ x 0 (t) − − − 0 ≤ t ≤ T, (4.44)

where the function x 0 (t) has resulted from the solution of Problem 4.4

at step 1.
Step 4. Check the relations of Eq. (4.38).
Step 5. Solve the initial-value problem of Eqs. (4.40) and (4.41) to find
the optimal control

u(t) for t > T . ¯

Proof of Basic Proposition To prove Proposition 4.1, first prove the

inequality

y 0 (t) ≤ J 1 (u) (4.45)

for any u(t) that satisfies the constraint

| u(t) |≤ U. (4.46)

y 0 (T ) for t > T .In
view of Eqs. (4.38) and (4.42), this would mean that the control u 0 (t)

Then prove that the control u(t) provides y 0 (t) ¯ ≡˜