tailieunhanh - Handbook of mathematics for engineers and scienteists part 84

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 84', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Nonlinear Systems of Ordinary Differential Equations 549 . Lyapunov function. Theorems of stability and instability. In the cases where the theorems of stability and instability by first approximation fail to resolve the issue of stability for a specific system of nonlinear differential equations more subtle methods must be used. Such methods are considered below. A Lyapunov function for system of equations is a differentiable function V V x1 . xn such that 1 2 V 0 if xk 0 V 0 if f V fk t fL 0 dt oxk xi xn 0 for t 0. Remark. The derivative with respect to t in the definition of a Lyapunov function is taken along an integral curve of system . Theorem stability Lyapunov . Let system have the trivial solution x1 x2 xn 0. This solution is stable if there exists a Lyapunov function for the system. Theorem asymptotic stability Lyapunov . Let system have the trivial solution xi xn 0. This solution is asymptotically stable if there exists a Lyapunov function satisfying the additional condition d -ß 0 with xk e1 0 t e2 0 k i where e1 and z2 are any positive numbers. Example 2. Let us perform a stability analysis of the two-dimensional system x t -ay - x x y yt bx - yip x y where a 0 b 0 x y 0 and x y 0 and ÿ are continuous functions . A Lyapunov function will be sought in the form V Ax2 By2 where A and B are constants to be determined. The first condition characterizing a Lyapunov function will be satisfied automatically if A 0 and B 0 it will be shown later that these inequalities do hold . To verify the second condition let us compute the derivative dv av av _ r - r- fi x y f2 x yi - -2 Ax ay x x y 2By bx - yip x y dt dx dy 2 Bb - Aa xy - 2Ax2 x y - 2By2 x y . setting here A b 0 and B a 0 thus satisfying the first condition we obtain the inequality -2bx2 x y - 2ay2 x y 0. dt This means that the second condition characterizing a Lyapunov function is also met. Hence the trivial solution of the system in question is .

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