How to impose physically coherent initial conditions to a fractional system

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On the sliding‐mode control of fractional‐order nonlinear uncertain dynamics (2016, result): “The notion of pseudo-states was considered originally in [26], and used later in numerous publications, including [20] and [27–29].”
Fractional Calculus: Quo Vadimus? (Where are we Going?) (2015, background): “But, fractional model concept would not have been a little hastily created? For instance, a look at the literature indicates that the definition of these models using Caputo’s derivative was contested [109, 98].”
Fractional systems state space description: some wrong ideas and proposed solutions (2014, methodology): “It was demonstrated by Sabatier et al. (2008, 2010a) with a counter example, that Caputo’s definition does not permit one to obtain the exact system pseudo trajectories if usual initial conditions such as (3) are taken into account in description (2).”