solve an inequality with LMI approach

I would like to solve this inequality :

Q*A'+A*Q+L'*B'+B*L < 0

[Amit]

solve for what? What the variable, what is known?

[John D'Errico]

Obviously, the poster wants to solve for x. Oh, x is not in the equation? So we can choose x=0. Case closed.

Seriously, before you ask a question, think about what you mean by it. What are the unknowns here? What are the knowns? Are they arrays? vectors? Scalars? What size? What do you mean by "solve", as that is not a terribly meaningful thing in terms of an inequality, as there will generally be infinitely many solutions to such a problem.

Once you define your problem, it MIGHT be possible to help you, but until then, impossible. Even then, solving an inequality is, as I said, generally not a well-posed problem.

[haifa]

the variables are the two matrix Q and L: Q: symetric matrix >0 L < 0

[Matt J]

And the inequality < 0 is element-wise, or is it the positive semi-definite ordering?

[haifa]

Q :is a symetric and semi definite positive L: <0

[Matt J]

Does L<0 mean that L(i,j)<0 for all i,j ?

[haifa]

no,L is a negativ matrix

Answer 1

[Johan Löfberg]

With the MATLAB Toolbox YALMIP, and some SDP solver installed (such as SDPT3, SeDuMi, Mosek etc) it would be (you have not clearly said which variables are decision variables, I assume Q (psd) and L (arbitrary))

Q = sdpvar(n,n);
L = sdpvar(m,n,'full');
Constraints = [Q >=0, Q*A'+A*Q+L'*B'+B*L <= 0];
solvesdp(Constraints);

The model is ill-posed though as Q and L arbitrarily close to zero is feasible, which means trouble in practice. Dehomogenioze it, for instance

Constraints = [Q >=0, Q*A'+A*Q+L'*B'+B*L <= -eye(n)];

[haifa]

I try to find the matrix Q:symetric and semi definite positive, and the negativ matrix L, which satisfy this inequality: Q*A'+A*Q+L'*B'+B*L < 0 knowing that: A=[0 1;0 0];B=[0;1];

[haifa]

I have not this solver SDP

[Johan Löfberg]

You have to install a solver (and YALMIP).

Start here http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Main.HomePage