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

Abstract
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: IfFis independent oft, anduis a constant input, then every bounded solution converges to a unique equilibrium pointx^{\ast}under some natural conditions. In addition, ifFis a nonlinear functional with separate variables, then every solution converges tox^{ast}under the same conditions; IfF(x_t,{\cdot})andu(\cdot)are periodic and have the same period\omega, then, under certain natural conditions, there is a\omega-periodic solutionx^{\ast}(\cdot), and every solution converges to it if it is a uniquew-periodic solution.

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