On a boundary value problem of a singular discrete time system with a singular pencil

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In this article, we study a boundary value problem of a class of singular linear discrete time systems whose coefficients are non-square constant matrices or square with a matrix pencil which has an identically zero determinant. By taking into consideration that the relevant pencil is singular, we provide necessary and sufficient conditions for existence and uniqueness of solutions. In addition, a formula is provided for the case of unique solutions and optimal solutions are studied for the cases of no solutions and infinite many solutions. Finally, based on a singular discrete time real dynamical system, numerical examples are given to justify our theory.