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Suppose I have the following 1st order linear differential matrix equation:

$$\dot{X}(t) = \frac{1}{2}D(t)X(t) + \frac{1}{2}X(t)D(t)^T $$

  1. $D(t),X(t)\in \mathbb{R}^{n\times n}$
  2. $X$ is positive semidefinite (i.e., $X\succeq 0$).

We know the following standard semidefinite programming form:

enter image description here

My question is how to embed that differential equation into the SDP?

You might consider the following scenario:
1. The variable is evolving over time and $D(t)$ is the given input data changing over time.
2. $C, A_i$ are constant matrices.

I do not want to use discretization, i.e., $$\dot{X} = \frac{X(k+1)-X(k))}{\Delta t}$$

This is because such differential equation is rank preserving (see the following lemma) and I want the rank of $X$ nor to change over time.

(Optimization and dynamical systems, Helmke and Moore)

enter image description here

Is there any possible way or related papers (or article with similar flavors) about this topic?

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    Have you considered creating a large block matrix whose diagonals are $X$ at time steps 1, 2, 3, ...? – 2017-02-25
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    @BrianBorchers If time step $k$ is large, then the size of the block will become very large. And how to deal with $\dot{X}$? I am confused here. – 2017-02-25
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    What do you mean by 'embed the differential equation into the SDP'? What are you trying to do? – 2017-02-25
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    The equation $\dot{x} = x^3$ is rank preserving but not of the form given in the lemma. – 2017-02-25
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    @copper.hat I want put that differential equation into the constraint of SDP such that the variable $X$ evolves over time according to that differential equation (dynamic). – 2017-02-25
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    Sorry, I have no idea what that even means. – 2017-02-25
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    @copper.hat I am using that form in Lemma because in my research I derive a differential equation fitting that form. – 2017-02-25
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    The euler's method step $X(k+1)=X(k)+\Delta t (D(k)X(k)+X(k)D(k)^{T})/2$ is a linear equation in $X(k)$ and $X(k+1)$. If (for example) $X$ was of size 5 by 5, then after 100 time steps, you'd end up with a matrix of size 500 by 500, containing 100 blocks of size 5 by 5. You'd also have 25*100=2500 constraints. – 2017-02-25
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    @BrianBorchers However the rank of $X$ may not be preserved(please see the lemma in my question) under the Euler's method step. It is like the length of unit quaternion will not be preserved under such discretization, ex: $\dot{q} = \frac{1}{2}\Omega q $ where $\|q\|=1$. – 2017-02-25
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    SDP isn't the way to control the rank of a matrix in any case. If you can come up with a rank preserving integration scheme that has a linear relationship between X(k) and X(k+1) then you can use the approach I described. I'm not aware of any such scheme for integrating ODE's. – 2017-02-25
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    _Chefd'Hotel, Christophe, et al. "[Regularizing flows for constrained matrix-valued images.](ftp://ftp-sop.inria.fr/odyssee/Publications/2003/chefdhotel-tschumperle-etal:03.pdf)" Journal of Mathematical Imaging and Vision 20.1 (2004): 147-162_ proposed a geometric numerical integrator that preserves the structure of the manifold of the symmetric matrices with fixed-rank (see in particular the Table 2). It may be useful for your purpose :) – 2017-02-26

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