Stability from the stabilization with general dissipativity constraint

The stabilization with a General Dissipativity Constraint (GDC) has been governed by the non-negativeness of ΔV(x, k) along the trajectories (i.e. ΔV(x, k)  0), in which ΔV(x, k)  0 is a storage function. We have stated and proved the state convergence with the GDC a previous work, but the stability has not been addressed thoroughly. In this paper, we analyze the stability that is obtained from the stabilization with the GDC in the context of Lyapunov stability, Lagrange stability and asymptotic stability. The GDC provides a type of stability that is similar to Lyapunov stability starting from a future time instant k∗ > 0. The GDC also provides a boundedness property that is similar to the Lagrange stability, but with a feasible condition. As a result, neither the Lyapunov stability nor the Lagrange uniform boundedness is obtained from the stabilization with the GDC.