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DEtools

  

matrix_riccati

  

solve a Matrix Riccati differential equation

 

Calling Sequence

Parameters

Description

Examples

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

Z' t=Zt.At.Zt+Zt.Kt+transposeKt.Zt

  

with

Zt0=Z0

  

where Z,A,K, and Z0 are n by n matrices. matrix_riccati returns the solution, Z.

• 

The command with(DEtools,matrix_riccati) allows the use of the abbreviated form of this command.

Examples

withDEtools:

The unknowns are xt,yt

ZMatrixxt,yt,yt,xt

Zxtytytxt

(1)

A simple example would be

AMatrixt,1,1,t

At11t

(2)

KMatrixc,0,0,c

Kc00c

(3)

The matrix with the initial values xt0 and yt0 is

Z0Matrix_C1,_C2,_C2,_C1

Z0_C1_C2_C2_C1

(4)

The system of equations represented by these matrices is thus

sysmapdiff,Z,t=Z.A.Z+Z.K+K%T.Z

sysⅆⅆtxtⅆⅆtytⅆⅆtytⅆⅆtxt=xtt+ytxt+xt+yttyt+2xtcxtt+ytytxt+yttxt+2ytcyttxtxt+xtt+ytyt+2ytcyttxtytxtt+ytxt2xtc

(5)

The two coupled odes are

ode1lhssys1,1=rhssys1,1

ode1ⅆⅆtxt=xtt+ytxt+xt+yttyt+2xtc

(6)

ode2lhssys1,2=rhssys1,2

ode2ⅆⅆtyt=xtt+ytytxt+yttxt+2ytc

(7)

The matrix solution to this matrix system of equations with initial conditions Z0 at t0=0 is computed as:

matrix_solmatrix_riccatiA,K,Z0,t=0:

Recalling the form of Z, the solution to the system of odes is constructed from matrix_sol as

solxt=matrix_sol1,1,yt=matrix_sol1,2

solxt=4ⅇtc2c22ⅇ2tc_C12ct+2ⅇ2tc_C22ctⅇ2tc_C12ⅇ2tc_C22+4_C1c2+_C12+_C224ⅇ2tc2_C12c2t2+4ⅇ2tc2_C22c2t2+4ⅇ2tc2_C12c24ⅇ2tc2_C12ct+4ⅇ2tc2_C22c24ⅇ2tc2_C22ct+16ⅇ2tc_C1c3t8ⅇ2tc_C12c2+4ⅇ2tc_C12ct8ⅇ2tc_C22c2+4ⅇ2tc_C22ct16ⅇ2tc_C2c3+ⅇ2tc2_C12+ⅇ2tc2_C228ⅇ2tc_C1c2+4_C12c2+4_C22c2+16_C2c3+16c42ⅇ2tc_C122ⅇ2tc_C22+8_C1c2+_C12+_C22,yt=8ⅇtc2c3ⅇ2tc_C12+ⅇ2tc_C22_C12_C222_C2c4ⅇ2tc2_C12c2t2+4ⅇ2tc2_C22c2t2+4ⅇ2tc2_C12c24ⅇ2tc2_C12ct+4ⅇ2tc2_C22c24ⅇ2tc2_C22ct+16ⅇ2tc_C1c3t8ⅇ2tc_C12c2+4ⅇ2tc_C12ct8ⅇ2tc_C22c2+4ⅇ2tc_C22ct16ⅇ2tc_C2c3+ⅇ2tc2_C12+ⅇ2tc2_C228ⅇ2tc_C1c2+4_C12c2+4_C22c2+16_C2c3+16c42ⅇ2tc_C122ⅇ2tc_C22+8_C1c2+_C12+_C22

(8)

This result can be verified with odetest

odetestsol,ode1,ode2

0,0

(9)

See Also

dsolve

odetest

riccati_system

riccatisol