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66 Chapter 3. Numeric solutions
Let for some r ∈ N denote Rr+ the cone of r-dimensional vectors with nonnegative coordinates. If we write C > 0 (C ≥ 0) for some matrix C ∈ Rn×m thismeansthatcij >0(cij ≥0)forall1≤i≤nand1≤j≤m.Ifamatrix C > 0, we call it positive and if C ≥ 0 we say that C is nonnegative, although this can be a bit misleading. Nonnegative here means that all elements of the considered matrix are nonnegative.
In this spirit we use an elementwise order relation, i.e. C > D iff C − D > 0. A real square matrix A ∈ Rn×n is called a Z-matrix if there exists s ∈ R and C ∈ Rn×n, C ≥ 0, such that A = sIn − C, where In denotes the n-dimensional unit matrix. A Z-matrix is a monotonic matrix in the order cone Rn+ called an M-matrix if s > ρ(C), where ρ : Rn×n → R+ denotes the spectral radius of the matrix. If s = ρ then A is called a singular M-matrix.
We recall the following results on Z-matrices being M-matrices (see [BeNeSt], [BePl])
Theorem 3.1.1. For a Z-matrix A, the following are equivalent: (i) A is an M-matrix.
(ii) A−1 ≥ 0.
(iii) Av > 0 for some vector v > 0.
(iv) All eigenvalues of A have positive real parts, i.e. −A is stable.
Notice that from (ii) we can infer that if Av > 0 for some vector v ∈ Rn then v > 0. In this way we apply (ii) in the subsequent section. Furthermore we have from [GuLa]
Theorem 3.1.2. Let A ∈ Rn×n be an M-matrix. If the elements of B ∈ Rn×n satisfy the relations
bii ≥aii, aij ≤bij ≤0, i̸=j, i,j =1,...,n, then B is also an M-matrix.
If the dynamical system (3.1) has an initial value x0 ∈ Rn+ and the control u only takes values in Rmi for i = 1,2, when is the state also nonnegative,
i+
i.e. x ∈ Rn+? This question is answered by a theorem wich can be found e.g. in [Za] or [FaRi] and is adapted for the presented situation:
Theorem 3.1.3. System (3.1) is positive if and only if B1 ≥ 0 and B2 ≥ 0 and the matrix −A is a Z-matrix.