Control of linear dynamical systems by time transformations
Consider a linear integral system \(S\) described by the equation \[ x(t)=x(0)+\int_0^tAx(s)\,ds+B\gamma (t),\quad t\geq 0,\tag{1} \] where \(x(0)\in{\mathbb{R}}^m\), \(A\) and \(B\) are \(m\times m\) and \(m\times d\) constant matrices, resp., \(\gamma :{\mathbb{R}}_+\rightarrow{\mathbb{R}}^d\) is a measurable bounded function called a configuration, and \(x :{\mathbb{R}}_+\rightarrow{\mathbb{R}}^m\) is an unknown function which describes the system \(S\) at time \(t>0\). Denote a solution of (1) as \(x(t,x,\gamma )\). Define a time transformation on the interval \([0, T]\) as a continuous strictly incremental function \(\lambda:\,[0, T]\rightarrow [0, T]\) such that \(\lambda (0)=0\) and \(\lambda (T)=T\). Denote by \(\Lambda_T\) the set of all time transformations on the interval \([0, T]\). \(\Lambda_T\)-control of the system \(S\) at zero on the interval \([0,T]\) with given \(\gamma\) is defined by the existence of a neighborhood \(U\) of \(0\in{\mathbb{R}}^m\) such that, for any \(x\in U\), there is a \(\lambda\in\Lambda_T\) satisfying the relation \(x(T,x,\gamma (\lambda (t))) =x(T,0,\gamma )\). Denote further by \(\sum_m^d\) the set of piecewise constant functions from \({\mathbb{R}}_+\) into \({\mathbb{R}}^d\) with \(m\) jumps. The main result is the proven equivalence among three controllability conditions (i)--(iii): (i) \(\operatorname{Rank}[AB,\dotsc,A^mB]=m\); (ii) there exist \(T>0\) and a bounded measurable configuration \(\gamma\) which admits the \(\Lambda_T\)-control of the system \(S\); (iii) there exists a configuration \(\gamma\in\sum_m^d\) for every \(T>0\) which admits a \(\Lambda_T\)-control of system \(S\). An illustrative example is supplied.
scientific article; zbMATH DE number 6128306