Abstract
This chapter describes the method of cone-valued Lyapunov functions. The technique of employing vector Lyapunov functions and the theory of vector differential inequalities offers a great deal of flexibility in studying nonlinear systems. However, for the success of this method, it is required that the comparison system be quasimonotone nondecreasing. In the case of linear comparison system, this requirement implies that the nondiagonal elements of the comparison matrix be all nonnegative. The limitation of the application potential of this general and effective method is because of the fact that comparison systems may have a desired property like positivity or stability of solutions without being quasimonotone. It was observed that this difficulty is because of the choice of the cone relative to the comparison system, namely Rn+, the cone of nonnegative elements of Rn and that a possible approach to overcome this limitation is to choose an appropriate cone other than Rn+. The chapter presents this idea and by developing the theory of differential inequalities through cones, the method of cone-valued Lyapunov functions is shown to be a useful tool in applications.
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