The Solvability of the Initial Problem for a Degenerate Linear Hybrid System With Variable Coefficients

We consider a linear hybrid systemwith variable coefficients and known mode switching moments under the assumption that matrices at the derivative of the desired vector function are identically degenerate. We obtain the necessary and sufficient conditions for the existence of a piecewise smooth solution (either continuous or not in its definition domain) for the initial problem. We study an equivalent form of a nonstationary system of linear ordinary differential equations that is not resolved with respect to the derivative and is identically degenerate in its definition domain. We propose a constructive algorithm for obtaining such a form even if the rank of the matrix at the derivative is not constant.