jacobrhcp
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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?
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?
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