Hello! I want to solve a system of Differential equations involving matrix of Unknowns. Can I use ode45?

Solve the following system of ODE's

\[
d/dt Y(t)=-(H(t))^{T}Y(t)
d/dt V(t)=-Y(t)^{T}Q^{T}QY(t)
\]

with the terminal conditions Y(T)=I_{2} and V(T)=0. Y, V, H and Q are 2*2 matrices.

In fact

\[
H(t)=H-Q^{T}Q\psi\left(T-t, R+M\right)
\]

where

$\psi\left(T-t, R+M\right)$

and

   $\phi\left(T-t, R+M\right)$

solve the following system of ODE's for

$\tau=T-t$

\

[
\frac{d\phi}{d\tau}=Tr[\beta Q^{T}Q\psi\left(\tau, R+M\right)]
\]
\[
\frac{d\psi}{d\tau}=\psi\left(\tau, R+M\right)H+H^{T}\psi\left(\tau, R+M\right)-2\psi\left(\tau, R+M\right)Q^{T}Q\psi\left(\tau, R+M\right)+R+M
\]
with the INITIAL conditions $\phi\left(0, R+M\right)=0$ and $\psi\left(0, R+M\right)=0$. $\psi\left(\tau, R+M\right)$ is a $2*2$ matrix and $\phi\left(\tau, R+M\right)$ is a scalar.  The $T$ in the superscript is transpose.

\section{Numerical Experiments}
For all experiments below
$T =15$; $\beta=3$ 
\[
R=
  \begin{bmatrix}
    1 & 0 \\
    0 & 0 
  \end{bmatrix}
\]
\[
M=
  \begin{bmatrix}
    0 & 0 \\
    0 & 1 
  \end{bmatrix}
\]
\[
H=
  \begin{bmatrix}
    -0.5 & 0.4 \\
    0.007 & -0.008 
  \end{bmatrix}
\]
\[
Q=
  \begin{bmatrix}
    0.06 & -0.0006 \\
    -0.06 & 0.006 
  \end{bmatrix}
\]

[RB]

Dear Carl,

Thanks for replying. Apologies for the editing. I have the terminal values but the equation is involving 2*2 matrices.

Solve the following system of ODE's

d/dt Y(t)=-(H(t))^{'}Y(t) 
d/dt V(t)=-Y(t)^{'}Q^{'}QY(t)

with the terminal conditions Y(T)=I_{2} and V(T)=0. Y, V, H and Q are 2*2 matrices and ' stands for transpose.

Also

H(t)=H-Q^{'}Q*W(T-t)

where W(T-t)=W(tau) is determined by the matrix Ricatti equation

dW/dtau = W(tau)H+H^{'}W(tau)-2W(tau)Q^{'}QW(tau)+I_{2}

such that at tau=0

W(0)=0.

I am badly stuck up.

I am unable to use ode45 since I want matrix result.

It would be great if you can give a suggestion.

\[
H=
  \begin{bmatrix}
    -0.5 & 0.4 \\
    0.007 & -0.008 
  \end{bmatrix}
\]
\[
Q=
  \begin{bmatrix}
    0.06 & -0.0006 \\
    -0.06 & 0.006 
  \end{bmatrix}
\]
Thanks.
Actually I am solving a system of matrix odes where dx/dt, dy/dt, dz/dt are all 2 by 2 matrices and I need to get X,Y,Z which are again 2 by 2 matrices. My code is
function [dXdt, dYdt, dVdt] = mwish7(t, X, Y, V, A, B, R) X = reshape(X, size(A)); Y = reshape(Y, size(A)); V = reshape(V, size(A)); dXdt = -A.'*X - X*A + 2*X*(B.'*B)*X - R; dYdt = -(A-(B.'*B)*X).'*Y; dVdt = -Y.'*(B.'*B)*Y; dXdt = dXdt(:); dYdt = dYdt(:); dVdt = dVdt(:); end
and the ode45 program I use is
A=[-0.5, 0.4; 0.007, -0.008]; B=[0.06 -0.0006; -0.06, 0.006]; R = [1 0; 0 1]; X0 = [0, 0; 0, 0]; Y0 = [1 0; 0 1]; V0 = [0, 0; 0, 0]; [T X Y V] = ode45(@(t,X,Y,V)mwish7(t, X, Y, V, A, B, R), [15 0], X0, Y0, V0)
I get the error: NOT ENOUGH INPUT ARGUMENTS.

Regards, RB

Answer 1

[Carl]

I'm unable to understand much of your question, due to the formatting. However, MATLAB, and ode45 specifically, can be used to solve a system of ODEs. See the following page, under the "Systems of ODEs" heading:

http://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html

And the following section in the documentation for ode45:

http://www.mathworks.com/help/matlab/ref/ode45.html#input_argument_odefun

Basically, you can structure the odefun input function to encode the system of equations you would like to solve for.