# Use symbolic variable for lyapunov function

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### Accepted Answer

Sam Chak
on 9 Mar 2022

If you are writing for a journal paper or a thesis, the following explanation might be helpful.

Let , , and .

There are a few ways to solve this symbolically.

syms a b c p11 p12 p22 p23 p33 p31

eqns = [1 - 2*a*p12 == 0, - a*p22 - b*p12 + c*p31 + p11 == 0, 1 - 2*b*p22 + 2*c*p23 + 2*p12 == 0, - b*p23 - c*p23 + c*p33 + p31 == 0, 1 - 2*c*p33 == 0, - a*p23 - c*p31 == 0];

S = solve(eqns);

sol = [S.p11; S.p12; S.p22; S.p23; S.p33; S.p31]

Result:

The result has been verified numerically:

clear all; clc

A = [0 1 0; -1 -2 0; 0 1 -1]

Q = eye(3)

P = lyap(A', Q)

A'*P + P*A

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