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Lyapunov's Second Theorem

Lyapunov's second theorem states that if all the eigenvalues of a real matrix A have negative real parts, then to an arbitrary negative definite quadratic form (x,Wx) with x=x(t) there corresponds a positive definite quadratic form (x,Vx) such that if one takes

(dx)/(dt)=Ax,

then (x,Vx) and (x,Wx) satisfy

d/(dt)(x,Vx)=(x,Wx).

See also

Lyapunov Equation, Lyapunov's First Theorem

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1122, 2000.

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Lyapunov's Second Theorem

Cite this as:

Weisstein, Eric W. "Lyapunov's Second Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LyapunovsSecondTheorem.html

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