Lyapunov Quadratic Check

Problem

For V(x) = x^T P x with P = [[p, q], [q, r]] and the linear system x' = A x with A = [[a, b], [c, d]], the derivative along trajectories is V-dot = x^T Q x with Q = P*A + A^T*P.

Return [q11, q12, q22, is_negative_definite], where Q = [[q11, q12], [q12, q22]]. A 2x2 symmetric matrix is negative definite when q11 < 0 and q11*q22 - q12^2 > 0.

Hint

A negative definite Q proves asymptotic stability of the origin.

Test cases

case 1

in: [0,1,-2,-3,1,0,1]

expected: [0,-1,-6,false]

case 2

in: [0,1,-2,-3,3,1,1]

expected: [-4,-2,-4,true]

case 3

in: [1,0,0,1,1,0,1]

expected: [2,0,2,false]

case 4

in: [-1,0,0,-2,1,0,1]

expected: [-2,0,-4,true]

case 5

in: [0,1,1,0,1,0,1]

expected: [0,2,0,false]

Starter codePython

def lyapunov_check(a, b, c, d, p, q, r):
    # Return [q11, q12, q22, flag]
    # Your code here
    pass

5 test cases · run your code against them in the editor