Invariance and monotonicity of nonlinear systems on time scales

Invariance of a subset of the state space with respect to a control system on an arbitrary time scale is studied. This includes in particular continuous-time and discrete-time systems. The invariance is characterized with the aid of natural operators associated to the system: shifts and skew derivations. For different time scales the criteria take different forms. Then for polynomial systems algebraic criteria of invariance based on Positivstellensatz from real algebraic geometry are developed. A relation between invariance and monotonicity is exhibited and the results on invariance are used to characterize monotonicity.