Proof Define v by V(x(0)= p(lr(t)l)dt where P: [ 0, oo)H0, oo) is positive for positive arguments and continuously differen- tiable. If V is correctly defined and differentiable, differentiation of v((t)) with respect to t at t=0 yields v(x(O)a(x(O)=-p(x(0)2) which proves the theorem. To make the integral convergent and continuously differen- tiable, it is sufficient to make p(y) converging to zero quickly enough as y-0 For the case of a linear system, a classical lyapunov theorem shows that local stability of equilibrium to =0 implies existence of a strict Lyapunov function which is a positive definite quadratic form Theorem 6.5 Ifa: R"H+R is defined by (z)=A where A is a given n-by-n matrin, then equilibrium Io=0 of(6.1)is locally asymptotically stable if and only if there erists a matric Q=Q>0 such that, for v(r)=i'QD VV(2)A=-22 6.1.3 Globally asymptotically stable equilibria Here we consider the case when a: R"bR in defined for all vectors. An equilibrium Co of (6.1) is called globally asymptotically stable if it is locally stable and every solution of(6.1) converges to o as t→∞ Theorem 6.6 If function V: R"HR has a unique minimum at io, is strictly mono- tonically decreasing along every trajectory of (6. 1)except r(t)=io, and has bounded level sets then io is a globally asymptotically stable equilibrium of (6.1) The proof of the theorem follows the lines of the proof of Theorem 6.4. Note that the assumption that the level sets of V are bounded is critically important: without it, some solutions of(6.1)may converge to infinity instead of To4 Proof Define V by � V (x(0)) = �(|x(t)| 2 )dt, 0 where � : [0,→) �� [0,→) is positive for positive arguments and continuously differen￾tiable. If V is correctly defined and differentiable, differentiation of V (x(t)) with respect to t at t = 0 yields ∈V (x(0))a(x(0)) = −�(|x(0)| 2 ), which proves the theorem. To make the integral convergent and continuously differen￾tiable, it is sufficient to make �(y) converging to zero quickly enough as y � 0. For the case of a linear system, a classical Lyapunov theorem shows that local stability of equilibrium x¯0 = 0 implies existence of a strict Lyapunov function which is a positive definite quadratic form. Theorem 6.5 If a : Rn �� Rn is defined by a(¯x) = Ax¯ where A is a given n- by-n matrix, then equilibrium x¯0 = 0 of (6.1) is locally asymptotically stable if and only if there exists a matrix Q = Q∗ > 0 such that, for V (¯x) = ¯x x, ∗ Q¯ ∈V (¯x)Ax¯ = −|x¯| 2 . 6.1.3 Globally asymptotically stable equilibria Here we consider the case when a : Rn �� Rn in defined for all vectors. An equilibrium x¯0 of (6.1) is called globally asymptotically stable if it is locally stable and every solution of (6.1) converges to x¯0 as t � →. Theorem 6.6 If function V : R ¯ n �� R has a unique minimum at x0, is strictly mono￾tonically decreasing along every trajectory of (6.1)except x(t) ≥ x¯0, and has bounded level sets then x¯0 is a globally asymptotically stable equilibrium of (6.1). The proof of the theorem follows the lines of the proof of Theorem 6.4. Note that the assumption that the level sets of V are bounded is critically important: without it, some solutions of (6.1) may converge to infinity instead of x¯0