Transcribed TextTranscribed Text

#1 Consider the nonlinear scalar differential equation x'(t)= ax(t) +f(t,x(t)),x(to)=x = , where a is real and constant and the function f is continuous with f(t,0) = 0. Assume that there is a constant > such that f (t.x)/<1/2). Use the Lyapunov function V(x==x, = to show that a) the zero solution of (1) is stable provided that a+à = 0. b) the zero solution of (1) is uniformly asymptotically stable (UAS) provided a + > < - for some positive constant a. c) Use part b) to show that the zero solution of = + 1 is (UAS). #2 Find a Lyapunov function to show the zero solution of x' = - x + y - x(x² + y2) y' = - 2y + y(x2 + y2) is (U.A.S). #3 Find a Lyapunov function to show the zero solution of x = y = Y==r3-y3 = - is stable. Hint: try V(x,y) = a.4 + by2 and then chose a and b, accordingly. #4 Show the zero solution of x' = 2xy + x³ y =-x²+y5 = - is unstable. #5 Consider = - where f is continuously differentiable. Show that a) if f(x,y) > 0, then the origin is A.S, b) whereas if f(x,y) < 0 the origin is unstable. Hint: Use appropriate polynomial Lyapunov function. #5 Consider the nonhomogenous system I' (t) = A(t):(t)+B(t), = Were A(t) and B(t) are continuous on 0 < to < t < 00, prove that if all solutions are bounded, then they are stable.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Advanced Math Problems
    $40.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Advanced Math Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats