OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 140

These relations imply the inequalities

| x |≤| y |+| ξ | , | y |≤| x |+| ξ | . (4.78)

Use these inequalities and the estimate of Eq. (4.76) for | ξ | to obtain

U
k ,

U
k ,

| x(t) |≤| y(t) |+ | y(t) |≤| x(t) |+ (4.79)

where the functions x(t) and y(t) define the solution of the system of
Eqs. (4.13) and (4.14) subject to the initial conditions of Eq. (4.15) and the
control u(t) . The inequalities of Eq. (4.79) imply the relations

U
k ,

U
k .
(4.80)

max
t ∈ [0 , ∞ ) |

x(t) |≤ max

t ∈ [0 , ∞ ) |

y(t) |+ max

t ∈ [0 , ∞ ) |

y(t) |≤ max

t ∈ [0 , ∞ ) |

x(t) |+

The behavior of the variable x in this solution coincides with the behavior
of the variable x in the solution of Eq. (4.28) subject to the initial conditions
of Eq. (4.29) for the same control u(t) . This follows from the fact that Eqs.
(4.13) and (4.28), as well as the initial conditions for x in Eqs. (4.15) and
(4.29), coincide. If the definitions of Eqs. (4.17) and (4.22) are utilized
for the criteria J 1 (u) and

J 1 (u) , then the relations of Eq. (4.80) can be

represented in the form

U
k ,

J 1 (u)

≤ +

J 1 (u) (4.81)

U
k .

J 1 (u) ≤ J 1 (u) + (4.82)

u 0 (the optimal control for Problem 4.2) into the inequality
of Eq. (4.81) to obtain

Substitute u =

U
k .

J 1 (u 0 )

≤ J 1 (u 0 ) + (4.83)

Since u 0 is the optimal control for Problem 4.4, it follows that ˜

J 1 ( J 1 (u 0 ).

˜ ≤

u 0 ) (4.84)

The inequalities of Eqs. (4.83) and (4.84) imply the relation

U
k .

J 1 (

u 0 ) ˜ − J 1 (u 0 ) ≤ (4.85)