How to solve the Lyapunov equation with unknowns

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Abdelkader Hd
Abdelkader Hd 2022-7-25
编辑： Torsten 2022-7-26
I'd like to ask those with unknowns Lyapunov What function does the equation use
The original equation is MV+VM‘=-D,
The matrix code is like this , There is only one unknown ：
M=[0 w1 0 0 0 0;-q1 -r1 0 0 -g1u -g1v;0 0 0 w2 0 0;0 0 -q2 -r2 -g2u -g2v;g1v 0 g2v 0 -k x;-g1u 0 -g2u 0 -x -k];
d=[0 r1*(2n1+1) 0 r2(2*n2+1) k k];
D=diag(d);
V=lyap(M,D)
The following error occurred after running , Only numeric matrices can be calculated ：
Misuse lyap (line 35)
The input arguments of the "lyap" command must be numeric arrays.
error Untitled (line 38)
V=lyap(M,D)
Thank you
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Abdelkader Hd
Abdelkader Hd 2022-7-26
编辑：Abdelkader Hd 2022-7-26
@Sam Chak well Thanks Sir for your response, I look to calculate logarithmic negativity "EN" and it's defined as a function of V (solution of Lyapunov equation). IF I give all numerical values of the parameters I don't find the variable for plot EN as a function of some parameters such as temperature coupling ..
Please see the picture below to understand what I'm looking for it
https://arxiv.org/pdf/1807.07158.pdf "link of the paper"
Thank you.
Sam Chak
Sam Chak 2022-7-26
编辑：Sam Chak 2022-7-26
I'm not good at magnomechanics, but it can clearly says that the elements of 𝒱are fully defined as function of u
and in Eq. (3), it is given that
.
So, maybe you can use ode45() to solve for . Then, you can find a steady-state 𝒱 when no longer change in time..

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Ivo Houtzager
Ivo Houtzager 2022-7-25
One inefficient way is too convert the Lyaponuv Matrix equation to linear system using the vectorization rule, and solve the linear system.
A = kron(eye(6),M) + kron(M,eye(6));
B = D(:);
X = A\B;
V = reshape(X,6,6);
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Torsten
Torsten 2022-7-26
编辑：Torsten 2022-7-26
@Sam Chak
If "\" and "kron" can deal with symbolic parameters, then in principle the problem can be solved using Ivo Houtzager 's suggestion. The inversion of a matrix does only use subdeterminants of the matrix itself - no roots are needed.
But even if it theoretically works: for a 36x36 or even 144x144 matrix, the expressions involved will become useless in my opinion.

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