Consider the linear system (27). a. Since (0, 0) is an asymptotically stable critical
Chapter 9, Problem 9(choose chapter or problem)
Consider the linear system (27). a. Since (0, 0) is an asymptotically stable critical point, show that a11 +a22 < 0 and a11a22 a12a21 > 0. (See of Section 9.1.) b. Construct a Liapunov function V( x, y) = Ax2+Bxy+Cy2 such that V is positive definite and V is negative definite. One way to ensure that V is negative definite is to choose A, B, and C so that V ( x, y) = x2 y2. Show that this leads to the result A = a2 21 + a2 22 + (a11a22 a12a21) 2 , B = a12a22 + a11a21 , C = a2 11 + a2 12 + (a11a22 a12a21) 2 , where = (a11 + a22)(a11a22 a12a21). c. Using the result of part a, show that A > 0, and then show (several steps of algebra are required) that 4AC B2 = _a2 11 + a2 12 + a2 21 + a2 22 _(a11a22 a12a21) + 2(a11a22 a12a21)2 2 > 0. Thus, by Theorem 9.6.4, V is positive definite.
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