Positional stabilization of two-dimensional bilinear systems with degenerate matrix of the bilinear form (Q5951173)

The authors consider a two-dimensional, single-input, bilinear system of the form \[ \dot x=Ax+uBx \] where the factorization \(B=bc^T\) holds, and the pairs \((A,b)\), \((A,c)\) are, respectively, controllable and observable. They distinguish between the cases \(c^Tb=0\) and \(c^Tb\neq 0\). In both cases, they give stabilization results by using methods of variable structure control, the discontinuity surface being defined by a linear map. The results are robust with respect to certain perturbations of the right-hand side of the system.