Some Comments on Positive Orthant Controllability of Bilinear Systems

Consider the bilinear system x = (A + uB)x, x ∈ Rⁿ, and u unrestricted. The system has the property that any solution $x(t)$ with x(0) 0 (i.e., all components of x nonnegative) will remain in the positive orthant, R_ + ⁿ = \ x ∈ Rⁿ |x 0\ for 0 t < ∞ if and only if B is diagonal and A = (aᵢⱼ ) has the property that aᵢⱼ 0 if i ≠ j. In this note the controllability of solutions from a point of R_ + ⁿ to another such point is studied. Some results are given for arbitrary $n > 0$ and detailed results are presented for the case $n = 2$.