Unless stated otherwise, we shall always assume that υ(t, 0) = 0 for all t ∈ R+ (resp., υ(0) = 0). Now, the function p(t) is defined as, Now, for the evaluation of the uninfected cells behavior, Eq. Let us suppose that φ (X, X) in (9.13) is negative definite. The theory of quadratic forms is used in the second-order conditions for a local optimum point in Section 4.4. By continuing you agree to the use of cookies. For Eq. We can now apply Theorem 2 to the system (4.1) and conclude that the system (4.1) is controllable ρ-stable for every ρ, 0 < ρ ≤ αm, if B is a control matrix for A. Here, one cannot check the signs of only leading principal minors, as was the case with the Sylvester criterion. SEE ALSO: Negative Definite Matrix, Positive Definite Matrix, Positive Semidefinite Matrix REFERENCES: Marcus, M. and Minc, H. A Survey of Matrix Theory and Matrix Inequalities. Since θ is bounded the stationary solution does not have zero flow as in (3.6), but instead one has the condition that P must be a periodic function of θ. Commuting. Let us put the result in a quadratic form in X modulo divergences. The function υ: R3 → R given by υ(x) = x21 + (x2 + x3)2 is positive semidefinite (but not positive definite). We let λ(x)≡12xT(A+AT)x and φ(α,x)≡αλ(x)−|BTx|, and first consider the case when (A + AT) is negative semidefinite. Theorem 2.3. NEGATIVE DEFINITE QUADRATIC FORMS The conditions for the quadratic form to be negative deﬁnite are similar, all the eigenvalues must be negative. and y~. Find the leading principal minors and check if the conditions for positive or negative definiteness are satisfied. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/S0076539208628222, URL: https://www.sciencedirect.com/science/article/pii/B9780128178010000120, URL: https://www.sciencedirect.com/science/article/pii/B978012374882900006X, URL: https://www.sciencedirect.com/science/article/pii/S0076539297800045, URL: https://www.sciencedirect.com/science/article/pii/S092465090680005X, URL: https://www.sciencedirect.com/science/article/pii/B9780123813756000048, URL: https://www.sciencedirect.com/science/article/pii/B9780444529657500118, URL: https://www.sciencedirect.com/science/article/pii/B9780123956514500374, URL: https://www.sciencedirect.com/science/article/pii/B9780128174616000056, URL: https://www.sciencedirect.com/science/article/pii/B0122274105001721, Convex Functions, Partial Orderings, and Statistical Applications, From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems, Non-Standard and Improperly Posed Problems, Introduction to Optimum Design (Third Edition), =0. A is negative definite if and only if Mk<0 for k odd and Mk>0 for k even, k=1 to n. A is negative semidefinite if and only if Mk<0 for k odd and Mk>0 for k even, k=1 to r

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