How to Determine Initial or Boundary Conditions for Linearized Systems?

[jacobrhcp]

Hi!

I have a system of four differential equations:

$\dot{x}(t)=Ax(t)+a$

where $A\in R^{4x4}$ and $a\in R^{4}$ are known and, $x(t)\in R^4 \forall t>0$

EDIT: the constraints I have on these differential equations are: $x_1(0)=x_{1,0}, x_2(0)=x_{2,0}, x_3(T)=x_{3,T}, x_4(T)=x_{4,T}$

I know I can decompose $A=MDM^{-1}$, where $D=diag(\lambda_1, ... , \lambda_4)$, and M is the matrix of eigenvectors (all of which I have computed the exact numbers of, important may or may not be that Re(lambda_i)>0 for i=1,2 and Re(lambda_i)<0 for i=3,4). By defining $y=M^{-1}x$, I can rewrite this system into:

$\dot{y}(t)=Dy+M^{-1}a$

Which I could solve had I known the initial conditions y_i(0) or the boundary conditions y_i(T). But I don't. I know (as this has been done before in a paper that does not elicit these technicalities) that I should somehow find y_i (0) or y_i (T) for each i=1,...,4. But how?

 

[chiro]

Hey jacobrhcp.

You need to provide some kind of constraint for the function or the initial condition to obtain the complete explicit definition of the system.

Even if you have to infer it in some way, it needs to be done if you want to calculate stuff and get numeric answers.

If this is an assignment then you should talk to your lecturer. If it's for a real world problem, then you need to think about any information that you can obtain that helps lead to using a reliable assumption that can give you a value or at least constrain it to a point where something reasonable can be used.

 

[jacobrhcp]

Yes! of course, you are very right, I am very sorry I did not write them down in the initial problem description, I've edited them in. Without these it's all nonsense. Though it's no assignment, it is a purely mathematical problem.

(in fact if you're interested: it is was mathematical model of the economy, that I linearized into a system like described above, and now I know it can be solved as some other people did it in papers and mumble words about linearization and then transformations, but I'm stuck on how to check and redo their work)