Qualitative analysis of nonlinear quasi-monotone dynamical systems described by functional-differential equations

This paper discusses properties related to the stability of a nonlinear quasi-monotone dynamical system described by a functionaldifferential equation
\dot{x} =F(x_t, t) + u(t)
. Specially, mathematical conditions which guarantee the same qualitative behavior inherent in a nonlinear off-diagonally monotone dynamical system
\dot{x}=f(x(t),t)+u(t)
are discussed. We first consider the basic properties of solutions: lower and upper bound preservation and ordering preservation of solutions. By using these properties, we estimate the trajectory. behavior by means of a partial ordering relation, and derive the following results: If
F
is independent of
t
, and
u
is a constant input, then every bounded solution converges to a unique equilibrium point
x^{\ast}
under some natural conditions. In addition, if
F
is a nonlinear functional with separate variables, then every solution converges to
x^{ast}
under the same conditions; If
F(x_t,{\cdot})
and
u(\cdot)
are periodic and have the same period
\omega
, then, under certain natural conditions, there is a
\omega
-periodic solution
x^{\ast}(\cdot)
, and every solution converges to it if it is a unique
w
-periodic solution.