Suppose I have the following 1st order linear differential matrix equation:
$$\dot{X}(t) = \frac{1}{2}D(t)X(t) + \frac{1}{2}X(t)D(t)^T $$
- $D(t),X(t)\in \mathbb{R}^{n\times n}$
- $X$ is positive semidefinite (i.e., $X\succeq 0$).
We know the following standard semidefinite programming form:
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)
Is there any possible way or related papers (or article with similar flavors) about this topic?