By Haddad, Wassim M.,

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**Example text**

26) is asymptotically stable. Hence, there exists 8 > 0 such that for all Xo E B8(0), x(t) ---+ 0 as t ---+ 00. 43) < t < T1 (xo), Xo E B8(0). 44), yields V(x(t)) < V(X(TI(XO)) + fd(X(T1 (xo))))e- C (t-71 (xo)) < V(X(T1(xo)))e- C (t- T l(X O )) < V (xo)e- CT1 (xo) e-c(t-Tl (xo)) = V(xo)e- ct , T1 (xo) < t < T2(XO), Xo E B8(0). 46) Recursively repeating the above arguments for t E (Tk (xo), Tk+ 1 (xo)], k = 3, 4, ... , it follows that V(x(t)) < V(xo)e- ct , t > 0, Xo E B8(0). 6 Two-mass system with constraint buffers.

Now, it follows that V(x) = 0, ~V(x) = (e 2 X E V, -1)mlm2(x2 2(ml + m2) X X E V. 56) V(x) = ". 55) is Lyapunov stable. 1. Considering the quadratic Lyapunov function candidate V (x) = X T Px, lImpulsive dynamical systems with fc(x) = Ax and fd(X) = (Ad - I)x are not linear. 60) establish asymptotic stability for linear state-dependent impulsive systems. These conditions are implied by P > 0, AJ'P + PAc < 0, and AIPAd - P < 0, which can be solved using a Linear Matrix Inequality (LMI) feasibility problem [27].

1). If and when the trajectory reaches a state Xl {). X(tl) satisfying (tl, xd E 8, then the state is instantaneously transferred to xi /: ,. 2). The trajectory x (t), tl < t < t2, is then given by "p(t, tl, xt), and so on. 2) is left-continuous, that is, it is continuous everywhere except at the resetting times tk, and Xk /: ,. 3) xt {). 4) e-O+ e-O+ for k = 1,2, .... To ensure the well-posedness of the resetting times we make the following additional assumptions: AI. If (t, x(t)) E 8\8, then there exists o < fJ < E, "p(t+fJ,t,x(t)) E > 0 such that, for all ~8.