ON THE CONTINUOUS FINITE-TIME STABILIZATION OF THE DOUBLE INTEGRATOR

Continuous finite-time stabilization is often treated under the analytical framework of homogeneity and has been frequently illustrated in the context of the feedback control of the double integrator. For such a simple system, the simplest considered continuous finite-time controller is composed of gained (proportional) exponentially weighted position and velocity error correction terms, with the exponential weights generally less than unity and constrained to satisfy a particular relation among them under homogeneity. What happens for less-than-unity exponential weights that do not satisfy such a homogeneity-based relation? Does the finite-time stabilization hold? Through a Lyapunov function--based study, we analyze and give more concrete answers to such questions than those partially provided by previous studies on the topic. We do find a more exhaustive spectrum of the exponential weights that give rise to finite-time stability of the trivial solution. Other types of stability properties are further found to take place for less-than-or-equal-to-unity exponential weights. Moreover, through complementary analysis, local or ultimate behavior of the system solutions is further characterized. The analytical findings are further illustrated through computer simulations.