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Your cost function can also be written as $$K = \int_0^{t_f} \left(\phi(t) - e^{-M^\top \tilde{D}\,M\,t} \hat{\phi}(0)\right)^\top \left(\phi(t) - e^{-M^\top \tilde{D}\,M\,t} \hat{\phi}(0)\right) dt.$$ When minimizing that cost function with respect to $\tilde{D}$ and $\hat{\phi}(0)$ it would be equivalent to minimizing the following cost function  K =...
I am a bit confused as to your characterization of constraints. Equation $(1)$ is not a constraint. It is the model that generated the time series data you are trying to fit. You then try to find the correct parameters $\tilde{D}$ that result in equation $(2)$ matching your time series as well as possible. I would formulate the problem as the following: ...