Canonical forms of linear discrete two-parameter systems (Q2703967)

Conditions under which a two-parameter system with infinite-order shift operators admits a canonical form, namely can be represented by an \(n\)-th order scalar equation with respect to a single variable, are provided. Let the sequence of linear operators (\(A_j\)), \(A_j:E \rightarrow E\) and the elements \( b_j \in E\), where \(E\) is a finite-dimensional Hilbert space over the complex numbers, be such that the Laurent series \(A(z)= \sum _{j=-\infty}^{+\infty} A_j z^j\) and \(b(z)= \sum _{j=-\infty}^{+\infty} b_j z^j\) are convergent in some unit annulus \(K\). The relations \((A_j)(s)= \sum_{j=-\infty}^{+\infty} A_j \varphi(s+j)\), \( (bv)(s)= \sum_{j=-\infty}^{+\infty} b_j v(s+j)\) determine linear bounded operators \( A : l^{\infty}(\mathbb{Z},E)\rightarrow l^{\infty}(\mathbb{Z},E)\) and \(b : l^{\infty}(\mathbb{Z},E)\rightarrow l^{\infty}(\mathbb{Z},E)\), where \(\mathbb{Z}\) is the set of integers, called shift operators, and the functions \( A(z)\) and \(b(z)\) are called their representations in the set of power series. A two parameter discrete-time system is given by the relation \(x(t+1,s)= \sum_{j=-\infty}^{+\infty} A_j x(t,s+j)+ \sum_{j=-\infty}^{+\infty} b_j u(t,s+j)\) for \(t=0,1, \) where \(x(t,s)\) is the unknown function with values in \(E\) and \(u(t,s)\) is the control with complex values. The notion of controllability is introduced as well as a group of transformation \(\mathbf L\) for such systems. When a system can be represented by an \(n\)-th order scalar equation with respect to a single variable it is called canonical. The possibility to reduce a system to canonical form using a transformation from \(\mathbf L\) is proved to be equivalent to the solvability of several problems. For instance any system that can be brought to canonical form using a transformation from \(\mathbf L\) is controllable. A system is controllable if and only if it is coefficient assignable. An example is given.