For the case of a linear system, however, local stability of equilibrium Io=0 implies existence of a Lyapunov function which is a positive definite quadratic form Theorem 6.2 Ifa: R"HR" is defined by a(i)=At here A is a given n-by-n matric, then equilibrium To=0 of(6. 1)is locally stable if and only if there erists a matrix Q=Q>0 such that V(r(t))=r(t)'Qa(t) is monotonically non-increasing along the solutions of(6.1) The proof of this theorem, which can be based on considering a Jordan form of A, is usually a part of a standard linear systems class 6.1.2 Locally asymptotically stable equilibria A point To is called a(locally) asymptotically stable equilibrium of (6. 1) if it is a stable quilibria, and, in addition, there exists eo>0 such that every solution of (6. 1)with (0)-iol Eo converges to To as t-0o Theorem 6.3 IfV: X HR is a continuous function such that V(x0)<V(z)V∈X/{fo}, and V(r(t)) is strictly monotonically decreasing for every solation of (6.1)encept r(t) To then To is a locally asymptotically stable equilibrium of(6.1) From Theorem 6.1, To is a locally stable equilibrium. It is sufficient to show that every solution =a(t) of(6. 1) starting sufficiently close to To will converge to Co as t-00. Assume the contrary. Then (t) is bounded, and hence will have at least one limit point I, which is not Io. In addition, the limit V of V(a(t)) will exist Consider a solution a*=a(t) starting from that point. By continuous dependence on initial conditions we conclude that V(,(t))=V is constant along this solution, which contradicts the assumptions A similar theorem deriving existence of a smooth Lyapunov function is also valid Theorem 6.4 If to is an asymptotically stable equilibrium of system(6. 1)where a: XH r" is a continuously differentiable function defined on an open subset X ofr" the erists a continuously differentiable function V: B(io)+R such that V(io)<v(i)for all≠ to and Vv(i)a(a)<0 VIE B(Co)/Col3 For the case of a linear system, however, local stability of equilibrium x¯0 = 0 implies existence of a Lyapunov function which is a positive definite quadratic form. Theorem 6.2 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 stable if and ∗ only if there exists a matrix Q = Q∗ > 0 such that V (x(t)) = x(t) Qx(t) is monotonically non-increasing along the solutions of (6.1). The proof of this theorem, which can be based on considering a Jordan form of A, is usually a part of a standard linear systems class. 6.1.2 Locally asymptotically stable equilibria A point x¯0 is called a (locally) asymptotically stable equilibrium of (6.1) if it is a stable equilibria, and, in addition, there exists e0 > 0 such that every solution of (6.1) with |x(0) − x¯0| < �0 converges to x¯0 as t � →. Theorem 6.3 If V : X �� R is a continuous function such that V (¯x0) < V (¯x) � x¯ ≤ X/{x¯0}, and V (x(t)) is strictly monotonically decreasing for every solution of (6.1) except x(t) ≥ x¯0 then x¯0 is a locally asymptotically stable equilibrium of (6.1). Proof From Theorem 6.1, x¯0 is a locally stable equilibrium. It is sufficient to show that every solution x = x(t) of (6.1) starting sufficiently close to x¯0 will converge to x¯0 as t � →. Assume the contrary. Then x(t) is bounded, and hence will have at x ¯ least one limit point ¯� which is not x¯0. In addition, the limit V of V (x(t)) will exist. Consider a solution x� = x�(t) starting from that point. By continuous dependence on ¯ initial conditions we conclude that V (x�(t)) = V is constant along this solution, which contradicts the assumptions. A similar theorem deriving existence of a smooth Lyapunov function is also valid. Theorem 6.4 If x¯0 is an asymptotically stable equilibrium of system (6.1) where a : X �� Rn is a continuously differentiable function defined on an open subset X of Rn then there exists a continuously differentiable function V : B�(¯x0) �� R such that V (¯x x 0) < V (¯) for all x¯ = ¯ ≡ x0 and ∈V (¯x)a(¯x) < 0 � x¯ ≤ B�(¯x0)/{x¯0}