  The paper is considered with the dissipative theory and feedback control
under the framework of dissipation with the supply rate is the inner product of
input u and the derivation of output y for the linear/nonlinear time-invariant
input affine system. A necessary and sufficient condition of dissipativity of
the class of systems is presented. Furthermore, we obtain necessary and
sufficient conditions of the negative imaginary property, input strictly
negative imaginary property and output strictly negative imaginary property for
the nonlinear input affine system, which can cover the case that the linear
time-invariant system in the literature. Besides, under given conditions, we
consider the equivalence between the negative imaginary property and
input-output stability. In other words, the system with negative imaginary
property may be input-output stable under some certain conditions; conversely,
the input-output stable system may also have negative imaginary property. Last
but not least, we consider the equivalence between the asymptotical stability
and strict dissipativity with the supply rate and some dissipative matrices are
given for a certain linear time-invariant system.
