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Reachable sets and controllability of bilinear time-invariant systems: A qualitative approach. (English) Zbl 0876.93011

The authors consider the bilinear dynamic system \[ \dot x= Ax+\sum^M_{i=1} (N_ix+ b_i)u_i,\quad x(0)= x_0,\quad t>0,\tag{1} \] where \(A\), \(N_i\) are \(n\times n\) matrices, \(b_i\in \mathbb{R}^M\), \(i=1,\dots,M\) and \(u= [u_1,\dots,u_M]\) is an \(M\)-dimensional control, \(u\in L^2(0,\infty;\mathbb{R}^M)\).
To analyze the reachability and controllability properties of the system (1), the authors propose a qualitative method. They use a constructive nonlinear decomposition technique in order to replace the study of the bilinear controlled system (1) by the study of \(2M+2\) associated uncontrolled linear systems \[ \begin{aligned} \dot x &=Ax\\ \dot x &= N_ix+ b_i\quad\text{or}\quad \dot x=-N_ix- b_i,\quad i=1,\dots,M.\end{aligned} \] Then, the standard techniques for linear polysystems can be applied. Some interesting properties for the homogeneous bilinear systems with commutative matrices were derived. Two examples are given to illustrate the results obtained.

MSC:

93B03 Attainable sets, reachability
93B05 Controllability
93B27 Geometric methods
93C99 Model systems in control theory
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