According to (14.1), the action trajectory must be
integrated to produce a state trajectory. In some cases, this
integration leads to a closed-form expression. For example, if the
system is a chain of integrators, then a polynomial expression can
easily be obtained for . For example, suppose
is a scalar
and
. If
and a constant action
is applied, then
. If
is a linear
system (which includes chains of integrators; recall the definition
from Section 13.2.2), then a closed-form expression for the
state trajectory can always be obtained. This is based on matrix
exponentials and is given in many control theory texts (e.g,
[192]).