Possible Close-form solution for 2 dimensional dΣdt=AΣ+ΣAT+BBTWhen A,B are 2x2 matrix and ∑(t) are PSD, can we expect a close form solution for the following ODE,dΣdt=AΣ+ΣAT+BBT.The problem is from solving the covariance of a time-invariant SDE x=Ax+Bdw.When x is in two dimensions, I guess there could be a close-form solution for ∑. But I am not sure how to solve it.

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Rey Mcmillan · Accepted Answer

Step 1Vectorize the equations=vec(Σ)&amp;nbsp;⟺&amp;nbsp;Σ=Mat(s)b=vec(BBT)s˙=(I⊗A+A⊗I)s&amp;nbsp;+&amp;nbsp;bs˙=Cs&amp;nbsp;+&amp;nbsp;bStep 2Substitute the dependent variable and solve the resulting ODE by inspectionx=s+C−1bx˙=s˙&amp;nbsp;=&amp;nbsp;Cx&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;x(t)=eCt&amp;nbsp;x(0)Recover the original variable and de-vectorize to matrix forms(t)=eCts(0)+(eCt−I)C−1b&amp;nbsp;&amp;nbsp;Σ(t)=Mat(s(t))

Adrien Ho · Accepted Answer

Step 1Notice thatddt[e−tAΣe−tAT]&amp;nbsp;=−e−tAAΣe−tAT+e−tAdΣdte−tAT−e−tAATΣe−tAT&amp;nbsp;&amp;nbsp;=e−tA(−AΣ+dΣdt−ΣAT)e−tAT&amp;nbsp;&amp;nbsp;=e−tABBTe−tAT.Step 2Hence,e−tAΣ(t)e−tAT−Σ(0)=∫0te−sABBTe−sAT,ds,i.e.,&amp;nbsp;Σ(t)=etAΣ(0)etAT+∫0te(t−s)ABBTe(t−s)AT,ds.

Possible Close-form solution for 2 dimensional \frac{d\Sigma}{dt} = \mathbf{A}\Sigma + \Sigma \ma

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