Determination of solutions to singular 2-D continuous-discrete linear systems with singular matrix pencils (Q1905694)

A method is presented for the determination of solutions to singular two-dimensional continuous-discrete linear systems described by the equation \(E\dot x(t,k+1)=Ax(t,k+1)+ B\dot x(t,k)+Cx(t,k)+f(t,x)\) with singular matrix pencils \(sE-A\). A set of admissible boundary conditions is established. It is shown that: (i) the equation always has a solution \(x(t,k)\) for any given matrices \(\{E,A,B,C\}\), acceptable input function \(f(t,k)\), and admissible boundary conditions: \(x(t,0)=y_1(t)\), \(t\in\mathbb{R}_+\), and \(x(0,k)=y_2(k)\), \(k\in\mathbb{Z}_+\). The components of \(x(t,k)\) which can be chosen arbitrary are characterized.