Theorem 6.1 to E X is a locally stable equilibrium of (6.1) if and only if there exist c>0 and a lower semicontinuous function V: Bio)H+R, defined on B2(0)={z:‖x-l‖l<e} and continuous at io, such that V(a(t)) is monotonically non-increasing along the solu tions of(6.1), and V(o<V(n)ViEBEo/ioJ Proof To prove that (ii)implies (i), define V(r)=inf(V(i)-v(io): I-Tol=r for r E(0, c). Since V is assumed lower semicontinuous, the infimum is actually a min- imum, and hence is strictly positive for all r E(0, c. On the other hand, since V continuous at o, V(r) converges to zero as r-0. Hence, for a given E>0, one can find d>0 such that V(min{e,c/2})>V(z)z:|-o<6. Hence a solution =a(t)of (6.1)with an initial condition such that l r(0)-iol<8(and hence V(z(o))<v(minE, c/2))cannot cross the sphere i-iol= mine, c/21 To prove that(i)implies(ii), define V by V(5)=suppl z(t)-Tol: t20,x(0)=I, a( satisfies(6.1)) Since by assumption, solutions starting close enough to io never leave a given disc cen- tered at io, V is well defined in a neigborhood Xo of o. Then, by its very definition V((t)) is not increasing for every solution of(6. 1)starting in Xo. Since V is a supremum, it is lower semicontinuous(actually, here we use the fact, not mentioned before, that if Tk=Tk(t) are solutions of( 6. 1)such that Tk (to)-i0o and Tk(t1)-ilo then there exists a solution of (6.1) with a(to)=i0 and a(t1)=ioo). Moreover, V is continuous at o because of stability of the equilibrium o One can ask whether existence of a Lyapunov function from a better class(say,con- tinuous functions) is possible. The answer, in general, is negative, as demonstrated by the following example Example 6.1 The equilibrium To=0 of the first order ODE Let a: R+R be defined erp(-1/i)sgn(i)sin2(E),I+0 a(T 7=0 Then a is arbitrary number of times differentiable and the equilibrium Io=0 of (6. 1)is locally stable. However, every continuous function V: R+R which does not increase along system trajectories will achieve a marimum at Io=0� 2 Theorem 6.1 x¯0 ≤ X is a locally stable equilibrium of (6.1) if and only if there exist c > 0 and a lower semicontinuous function V : Bc(¯x0) �� R, defined on Bc(¯x0) = {x¯ : ∞x − x¯0∞ < c} and continuous at x¯0, such that V (x(t)) is monotonically non-increasing along the solu￾tions of (6.1), and V (¯x0) < V (¯x) � x¯ ≤ Bc(¯x0)/{x¯0}. Proof To prove that (ii) implies (i), define V x) − V (¯ x − ¯ ˆ (r) = inf{V (¯ x0) : |¯ x0| = r for r ≤ (0, c). Since V is assumed lower semicontinuous, the infimum is actually a min￾imum, and hence is strictly positive for all r ≤ (0, c). On the other hand, since V is continuous at x¯0, Vˆ (r) converges to zero as r � 0. Hence, for a given � > 0, one can find � > 0 such that V x) � ¯ x − ¯ ˆ (min{�, c/2}) > V (¯ x : |¯ x0| < �. Hence a solution x = x(t) of (6.1) with an initial condition such that |x(0) − x¯0| < � (and hence V (x(0)) < Vˆ (min{�, c/2}) cannot cross the sphere |x¯ − x¯0| = min{�, c/2}. To prove that (i) implies (ii), define V by V (¯x) = sup{∞x(t) − x¯0∞ : t ∀ 0, x(0) = x, ¯ x(·) satisfies (6.1) }. (6.2) Since, by assumption, solutions starting close enough to x¯0 never leave a given disc cen￾tered at x¯0, V is well defined in a neigborhood X0 of x0. Then, by its very definition, V (x(t)) is not increasing for every solution of (6.1) starting in X0. Since V is a supremum, it is lower semicontinuous (actually, here we use the fact, not mentioned before, that if 0 � and xk(t1) � x¯1 xk = xk(t) are solutions of (6.1) such that xk(t0) � x¯ � then there exists 0 � and x(t1) = x¯1 a solution of (6.1) with x(t0) = x¯ �). Moreover, V is continuous at x0, because of stability of the equilibrium x0. One can ask whether existence of a Lyapunov function from a better class (say, con￾tinuous functions) is possible. The answer, in general, is negative, as demonstrated by the following example. Example 6.1 The equilibrium x¯0 = 0 of the first order ODE Let a : R �� R be defined by x2 exp(−1/¯ )sgn(¯x)sin x), x ≡ 2 (¯ ¯ = 0, a(¯x) = 0, x¯ = 0. Then a is arbitrary number of times differentialble and the equilibrium x¯0 = 0 of (6.1) is locally stable. However, every continuous function V : R �� R which does not increase along system trajectories will achieve a maximum at x¯0 = 0