HEAD INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 237

problems are stated as optimal control problems in which the HIC is to be
either minimized or constrained.
A comprehensive analysis of the optimal control problems with the HIC
functional has not yet been performed, although an optimal control has been
constructed in some special cases. For example, the solution is known for
the problem of the optimal deceleration of a point mass (particle) moving
along a straight line with a given initial velocity. The deceleration distance
is minimized with the condition that the HIC does not exceed a prescribed
admissible value. The parameter is assumed to be equal to the duration
of the deceleration pulse. This model allows one to evaluate the potentials
to reduce the risk of head injuries due to impacts by using helmets or
surface coatings against which the impacts occur. The analytical solution
of this problem is presented (without proof) by Okamoto et al. (1994) in
connection with the crashworthy design of a car hood to reduce the head
injury to a pedestrian who is impacted by the hood. Cheng et al (1999)
constructed a numerical solution of a similar problem in connection with
the limiting performance analysis of impact protection helmets.
Hutchinson, Kaiser, and Lankarani (1998) studied mathematical features
of the HIC for the case where the function a(t) in Eq. (7.1) is defined on a
finite time interval, vanishes at the ends of this interval, and is continuous
and piecewise differentiable, and the parameter coincides with the length
of the interval on which the function a(t) is defined. The class of functions
a(t) considered in the cited paper covers all cases that can be encountered
in practice. Therefore, the algorithms proposed there are suitable for the
determination of the upper bound for the HIC [i.e., the maximum value of
the functional of Eq. (7.1) over all ] when processing crash-test measure-
ment data. At the same time, for theoretical analyses of problems involving
the HIC, in particular optimal control problems, it is reasonable to extend
the class of functions a(t) to that of functions integrable (generally speak-
ing, in the improper sense) on a prescribed time interval. It will be shown
that the solution of optimal control problems with the HIC functional can
lead to functions a(t) tending to infinity at some time instants, the number
of such instants depending on the parameter .
The optimal deceleration law will be constructed here for the rectilin-
ear translational (without rotation) motion of a headform. In this case, the
headform can be modeled as a point mass. The deceleration distance will be
minimized subject to a constraint on the HIC functional. Analytical expres-
sions are obtained for the optimal control and the minimal deceleration
distance. The qualitative behavior of the optimal control is investigated as
a function of the mass of the headform, the initial velocity (impact velocity),
the maximum allowable value of the HIC, and the parameter occurring in