Controllability of Piecewise Systems With Noncommutative Coefficient Matrices and State-Dependent Delays

The article investigates the controllability of piecewise systems with non-commutative coefficient matrices and state-dependent delays (PSCMSD). First, an array of functions is constructed and appropriate formulations of the solution to PSCMSD are developed. Then, two necessary and sufficient conditions, specifically the Gram criterion and the rank criterion, are established to evaluate the controllability of linear PSCMSD. Moreover, a minimally normalized control function capable of steering the solution of linear PSCMSD to a desired final state is explored. Furthermore, the controllability of semilinear PSCMSD is discussed and two sufficient criteria are obtained by employing the Krasnoselskii's and Schauder's fixed point theories. Finally, an RLC circuit model illustrates the applicability of the results.