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By John G. Webster (Editor)

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As a simple example, diagonal systems x˙ i = ui xi , i = 1, . . , n, xi > 0 on the orthant Rn+ = {x ∈ Rn |xi > 0, i = 1, . . , n} can obviously be transformed by the substitution xi = exp(zi ) to the system of coordinate-translations z˙ i = ui . Controllability for systems of a more interesting nature that have positive orthants as their natural state spaces was studied by Boothby (28). Here the BLS is x˙ = Ax + uBx on Rn+ (17) under the hypothesis that the n × n matrix A is essentially positive: ai j > 0, i = j, written A > 0.

Qn ). Then if the numbers qi − q j are all distinct, if the elements of A satisfy Ai j = 0 for all i, j such that |i − j| = 1, and if A1n An1 < 0, then x˙ = (A + uB)x is controllable on Rn \0. Jurdjevic and Sallet (27) and others have extended the approach of Reference 24 to nonhomogeneous BLS like equation 3, in the m-input case. Positive Systems. There are several ways in which BLS models arise that involve positive variables. As a simple example, diagonal systems x˙ i = ui xi , i = 1, . .

For n ≥ 3 the puncture has negligible effect. In some applications, other unusual state spaces may be appropriate. There is an n-dimensional generalization of our scalar example, the diagonalizable BLS, which are (up to similarity) given by x˙ i = ui xi , i = 1, . . , n. If the initial state is on a coordinate half-axis, a quadrant of a coordinate plane, . . , or on one of the 2n orthants, the state must stay there forever. Other BLS that live this way on positive orthants occur rather often, and their controllability properties will be discussed in the “Positive Systems” section below.

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04.Automatic Control by John G. Webster (Editor)


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