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On the lumpability of linear evolution equations
Fatihcan M. Atay and Lavinia Roncoroni
We analyze the lumpability of linear differential equations on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction operator onto a smaller state space on which a self-contained dynamical description exists. We first consider systems whose evolution is described by bounded linear operators, and extend previous results by relaxing some of the hypotheses. The lumpability condition is then expressed as the invariance of the kernel of the reduction operator under the evolution operator. Next, as the main contribution of the paper, we consider dynamics defined by unbounded operators. We use methods from the theory of strongly continuous semigroups to obtain conditions on the reduction operator for lumpability. We indicate several applications to particular systems, including delay differential equations.