Minimal algebraic realization of bilinear discrete dynamical systems (Q1068030)

A bilinear discrete dynamical system is given by the equations \[ x(t+1)=Ax(t)+\sum^{p}_{i=1}u_ i(t)B_ ix(t)+Cu(t),\quad y(t)=Dx(t),\quad x(0)=x, \] where \(t=0,1,...\), \(x(t)\in R^ n\) is the state at t, \(u(t)\in R^ p\) is the control at t with i-th coordinates \(u_ i(t)\), \(y(t)\in R^ q\) is the output at t, A, \(B_ i\), C, D are fixed matrices. Let \(\lambda\) (x,u,t) be the state at t, with initial condition x and control sequence \(u=\{u(t)\}^{\infty}_{t=0}\). The system is called accessible from x if the linear span of the set of all states x' for which \(x'=\lambda (x,u,t)\) for some t and some control sequence u, is equal to \(R^ n\). The system is called observable if given distinct \(x',x''\in R^ n\) there is a control sequence u and t such that \(D\lambda\) (x',u,t)\(\neq D\lambda (x'',u,t)\). The system is called minimal (with respect to x) if it is accessible from x and observable. In the reviewed paper necessary and sufficient conditions are given for minimality of various classes of bilinear discrete dynamical systems. The ideas and methods of universal algebra are used in the proofs.