Hello everyone. Good news!!!

Finally I changed the candidate of my Lyapunov function, now it has 3 coefficients and I derived 9 equations to impose d/dt(V(x))<0. 

The equations are redundant, but I solved them with Matlab and I got many solutions (all three variables equal to the same value).

I think that the original problem I tried to solve has no solution such that all the variables are real and different from zero, that's why I chose to change Lyapunov candidate. 

Thank you very much for your help and your time. You're very very kind!

"For real solutions the DirectSearch approach could be OK but we must be sure not to obtain just an approximation of the null solution." 

Yes, that's why I have some doubts when there's a solution with very small values. 

I added the constraint because I want to know if there's at least one solution where none of the variables is zero.  

Is there a way to see if there's at least one solution where the variables are not zero?

And, is there a way to get them real? 

@tomleslie 

I thank you. 

When I type DirectSearch on the Search bar I get 'No matches found', but I followed the instructions in the ReadMe file that was attached to the zip of the package. I copied the .mla and .hdb files in /lib directory and I also run the command 

libname:=”C:\\DirectSearchLib”, libname; 
 

@vv 

Hi, thank you very much for your kind help.

I just wanted to know if it was 'solvable', because I'm looking for a Lyapunov function for a linear system and I chose one candidate that depends on 12 coefficients (a1,a2,b1...). I found the expression of d/dt(V(x)) and, in order to have it negative, I derived those 9 equations and so, if at least one solution exists, V(x) is a Lyapunov function for the system. 

Do you mean that the original system with 9 equations has infinite solutions? Because I was using command

solve(sys, {a1, a2, b1, b2, c1, c2, d1, d2, e1, e2, f1, f2},maxsols=1)

but the evaluation lasts too much time.

I'm doing as you suggested, but I want a solution where the variables are not zero. That's why I added the constraint that a1>0. With this additional constraint, the evaluation lasts for a long time, is it normal? 

@tomleslie 

Hi, thank you very much for your help. I downloaded Direct search optimization package and I copied files DirectSearch.mla and DirectSearch.hdb inside the directory C:\Program Files\Maple 2020\lib , but when I try to run the command

ans := DirectSearch:-SolveEquations(sys, evaluationlimit = 20000)

I get " SolveEquations is not a command in the DirectSearch package " .

I was wandering whether a solution exists, because actually I derived the system of equations in order to find a suitable Lyapunov function for my linear system and, with these equations, I am able to impose d/dt(V(x)) < 0. 

My only concern is that they are not null. They are quite 'small' but my requirement is just that they are not zero. 

Maybe, increasing or decreasing constant c, the values of the solutions become larger?