|
DEtools[matrix_riccati] - solve a Matrix Riccati differential equation
|
|
Calling Sequence
|
|
matrix_riccati(A, K, Z0, t)
matrix_riccati(A, K, Z0, t=t0)
matrix_riccati(A, K, t)
matrix_riccati(A, K, t=t0)
|
|
|
Parameters
|
|
|
A, K
|
-
|
Matrix coefficients of the Riccati Matrix Equation
|
|
Z0
|
-
|
Matrix representing the initial conditions for t=t0
|
|
t
|
-
|
independent variable
|
|
t0
|
-
|
position of initial conditions
|
|
|
|
|
|
Description
|
|
|
•
|
The Matrix Riccati differential equation is
|
|
•
|
The command with(DEtools,matrix_riccati) allows the use of the abbreviated form of this command.
|
|
|
|
Examples
|
|
|
>
|
|
The unknowns are 
|
>
|
![Z := Matrix([[x(t), y(t)], [y(t), -x(t)]])](/support/helpjp/helpview.aspx?si=8699/file01351/math136.png)
|
|
![Z := Matrix([[x(t), y(t)], [y(t), -x(t)]])](/support/helpjp/helpview.aspx?si=8699/file01351/math139.png)
| (1) |
A simple example would be
|
>
|
![A := Matrix([[-t, 1], [1, t]])](/support/helpjp/helpview.aspx?si=8699/file01351/math145.png)
|
|
![A := Matrix([[-t, 1], [1, t]])](/support/helpjp/helpview.aspx?si=8699/file01351/math148.png)
| (2) |
|
>
|
![K := Matrix([[c, 0], [0, c]])](/support/helpjp/helpview.aspx?si=8699/file01351/math152.png)
|
|
![K := Matrix([[c, 0], [0, c]])](/support/helpjp/helpview.aspx?si=8699/file01351/math155.png)
| (3) |
The matrix with the initial values ![x(t[0])](/support/helpjp/helpview.aspx?si=8699/file01351/math158.png) and ![y(t[0])](/support/helpjp/helpview.aspx?si=8699/file01351/math160.png) is
|
>
|
![Z0 := Matrix([[_C1, _C2], [_C2, -_C1]])](/support/helpjp/helpview.aspx?si=8699/file01351/math165.png)
|
|
![Z0 := Matrix([[_C1, _C2], [_C2, -_C1]])](/support/helpjp/helpview.aspx?si=8699/file01351/math168.png)
| (4) |
The system of equations represented by these matrices is thus
|
>
|
|
|
![sys := Matrix([[diff(x(t), t), diff(y(t), t)], [diff(y(t), t), -(diff(x(t), t))]]) = Matrix([[(-x(t)*t+y(t))*x(t)+(x(t)+y(t)*t)*y(t)+2*x(t)*c, (-x(t)*t+y(t))*y(t)-(x(t)+y(t)*t)*x(t)+2*y(t)*c], [(-y(t)*t-x(t))*x(t)+(-x(t)*t+y(t))*y(t)+2*y(t)*c, (-y(t)*t-x(t))*y(t)-(-x(t)*t+y(t))*x(t)-2*x(t)*c]])](/support/helpjp/helpview.aspx?si=8699/file01351/math177.png)
| (5) |
The two coupled odes are
|
>
|
![ode[1] := lhs(sys)[1, 1] = rhs(sys)[1, 1]](/support/helpjp/helpview.aspx?si=8699/file01351/math183.png)
|
|
![ode[1] := diff(x(t), t) = (-x(t)*t+y(t))*x(t)+(x(t)+y(t)*t)*y(t)+2*x(t)*c](/support/helpjp/helpview.aspx?si=8699/file01351/math186.png)
| (6) |
|
>
|
![ode[2] := lhs(sys)[1, 2] = rhs(sys)[1, 2]](/support/helpjp/helpview.aspx?si=8699/file01351/math190.png)
|
|
![ode[2] := diff(y(t), t) = (-x(t)*t+y(t))*y(t)-(x(t)+y(t)*t)*x(t)+2*y(t)*c](/support/helpjp/helpview.aspx?si=8699/file01351/math193.png)
| (7) |
The matrix solution to this matrix system of equations with initial conditions  at ![t[0] = 0](/support/helpjp/helpview.aspx?si=8699/file01351/math198.png) is computed as:
|
>
|
|
Recalling the form of  , the solution to the system of odes is constructed from  as
|
>
|
![sol := {x(t) = matrix_sol[1, 1], y(t) = matrix_sol[1, 2]}](/support/helpjp/helpview.aspx?si=8699/file01351/math213.png)
|
|
| (8) |
This result can be verified with odetest
|
>
|
![odetest(sol, [ode[1], ode[2]])](/support/helpjp/helpview.aspx?si=8699/file01351/math225.png)
|
|
![[0, 0]](/support/helpjp/helpview.aspx?si=8699/file01351/math228.png)
| (9) |
|
|
Download Help Document
Was this information helpful?
