The article of record as published may be located at http://dx.doi.org/10.1080/00036810108840984The method of vector Lyapunov functions to determine stability in dynamical systems requires that the comparison system be quasimonotone nondecreasing with respect to a cone contained in the nonnegative orthant. For linear comparison systems in Rn with real spectra, Heikkilﾨa solved the problem for n = 2 and gave necessary conditions for n > 2. We previously showed a su_cient condition for n > 2, and here, for systems with complex eigenvalues, we give conditions for which the problem reduces to the nonnegative inverse eigenvalue problem

Beaver, P.

Canright, David R.

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveFaculty and Researcher Publications Faculty and Researcher Publications2002The Quasimonotonicity of LinearDifferential Systems - The Complex SpectrumBeaver, P.Applicable Analysis / Volume 80, 127-131http://hdl.handle.net/10945/255341THE QUASIMONOTONICITY OF LINEARDIFFERENTIAL SYSTEMS—THE COMPLEXSPECTRUMCommunicated by P. BeaverP. BEAVERa, D. CANRIGHTbaDepartment of Mathematical SciencesUnited States Military AcademyWest Point, NY 10996, USAphilip-beaver@usma.edu;bDepartment of MathematicsNaval Postgraduate SchoolMonterey, CA 93943, USAdcanright@nps.navy.milAMS: 15A48, 34D20Abstract The method of vector Lyapunov functions to determine stability indynamical systems requires that the comparison system be quasimonotonenondecreasing with respect to a cone contained in the nonnegative orthant. For linearcomparison systems in Rn with real spectra, Heikkila¨ solved the problem for n = 2 andgave necessary conditions for n > 2. We previously showed a sufficient condition forn > 2, and here, for systems with complex eigenvalues, we give conditions for which theproblem reduces to the nonnegative inverse eigenvalue problem.KEY WORDS: Vector Lyapunov functions, cones, quasimonotonicity, dynamical systems.(Received for Publication 12 Sep. 2001)1. IntroductionThe technique of vector Lyapunov functions has proven very useful in analyzing thestability of differential systems (see, e.g., [9]). In this method, however, the comparisonsystem requires stability properties as well as quasimonotonicity relative to a cone (see[8]). For linear comparison systems, it is sufficient to consider square systems A(t) ∈Rn×n, t ≥ 0, and to find a cone contained in R n+ with respect to which the system isquasimonotone (see [1]).Heikkila¨ [6] showed that to solve the problem for linear comparison systems A ∈ Rn×n,it is necessary and sufficient to find a nonnegative, nonsingular matrix B ∈ Rn×n suchthat if C = B−1AB, then cij ≥ 0 for all i 6= j. In this case we say that A is cross-positiveon the proper, simplicial cone determined by the columns of B (see [10]) and that C isessentially nonnegative (see [11]).In [6], Heikkila¨ used Perron-Frobenius theory to show a necessary condition is for thematrix A to have a real eigenvalue with greatest real part and an associated nonnegativeeigenvector. For n = 2, Heikkila¨ showed that it is necessary and sufficient that A havetwo nonnegative eigenvectors. For n > 2 he further showed that it is sufficient for amatrix A with a real spectrum to have all of its eigenvectors nonnegative, and in [7],Ko¨ksal and Fausett extended this result to include generalized eigenvectors. In [1], weshowed that with n ≥ 2, for matrices with real spectra it is sufficient that this firsteigenvector be positive, and we discussed cases when it is nonnegative in terms of thereducibility of the matrix A.2 Quasimon. Of Lin. Sys.—Complex SpectrumThis problem has further applications to the positive orthant stabilizability problem(see [3]) and the “hit and hold” problem (see [2]) from control theory. In this paper weaddress the case of the general spectrum (when the eigenvalues of A are not strictly real)and we present conditions under which the problem reduces to the nonnegative inverseeigenvalue problem.2. The General SpectrumIn the case of the real spectrum, all of our solutions used the fact that the diagonal(or Jordan canonical) form of the matrix A is essentially nonnegative, since all of itsoff-diagonal entries are zero (or 1). When a matrix C with real spectrum is essentiallynonnegative, then there always exists an r such that C + rI is nonnegative. However,when A has complex eigenvalues, it is possible that it is never similar to an essentiallynonnegative matrix. For example, if the spectrum of A, σ(A) = {1, i,−i}, then no matrixB exists such that C = B−1AB is essentially nonnegative. This is because there is noreal r for which the shifted spectrum σ(C + rI) = {1 + r, i+ r,−i+ r} is the spectrumof a nonnegative matrix. (The spectrum σ = {1 + , i,−i} is in fact the spectrum of anonnegative matrix for any  > 0.)It is clear that in order for a matrix A to be quasimonotone nondecreasing withrespect to a nonnegative cone, it is necessary that, for some real r, the shifted spectrumσ(A + rI) be the spectrum of some nonnegative matrix, i.e., the shifted spectrum of Amust solve the nonnegative inverse eigenvalue problem. Inverse eigenvalue problems arewell-studied problems in linear algebra (see [5]) and perhaps the most important unsolvedproblem of this type is the nonnegative inverse eigenvalue problem, which asks when aset of numbers is the spectrum of a nonnegative matrix. Although the problem has beenstudied extensively, in general it remains unsolved (see, for example, [4]).We offer a solution to our problem when the shifted spectrum of A (for some realr) solves the irreducible nonnegative inverse eigenvalue problem, in that σ(A + rI) isthe spectrum of an irreducible nonnegative matrix An. (The case where the shiftedspectrum of A can be decomposed into subsets, each of which solves the irreduciblenonnegative inverse eigenvalue problem individually, requires that we extend the theory ofthis paper using techniques similar to those shown in [1] for reducible matrices. However,unfortunately no general theory exists based strictly on the spectrum of A for suchmatrices.)If there exists a B ≥ 0 such that Ae = B−1AB is essentially nonnegative and ir-reducible, then Perron-Frobenius theory tells us that Ae has a real eigenvalue λ1 withλ1 > Re(λi) for i = 2, ..., n and an associated positive eigenvector y1 (since An = Ae+rIis nonnegative and irreducible). Therefore, it is necessary for A to have an eigenvalue λ1with λ1 > Re(λi) for i = 2, ..., n and an associated positive eigenvector x1 since x1 = By1with y1 > 0, B ≥ 0, and B nonsingular. If A and Ae are similar, we show that it issufficient that A have a positive first eigenvector in order for the change-of-basis matrixB to be nonnegative, so that A is quasimonotone with respect to a nonnegative cone.We note the similarity between this result and the result in [1] for matrices with realspectra, where it was also sufficient that A have a first eigenvector x1 > 0.P. Beaver & D. Canright 33. A Theorem For The General SpectrumTheorem. Let A ∈ Rnxn for n > 2. In order for A to be quasimonotone nondecreasingwith respect to a nonnegative cone, it is necessary that for some real r the shifted spectrumσ(A + rI) solve the nonnegative inverse eigenvalue problem, and that A have a firsteigenvalue λ1 where λ1 > Re(λi) for i = 2, ..., n with an associated nonnegative firsteigenvector x1 ≥ 0. Furthermore, it is sufficient that A be similar to an irreducibleessentially nonnegative matrix Ae and that A have a positive first eigenvector x1 > 0.Proof. The necessary conditions follow from the previous discussion, and we note thatfor matrices with real spectra, the requirement on the shifted spectrum is always triviallysatisfied. To show the sufficient conditions, let Ae be similar to A where Ae is irreducibleand essentially nonnegative, so that σ(A+ rI) solves the irreducible nonnegative inverseeigenvalue problem. Furthermore, let A have a positive first eigenvector x1 > 0 associatedwith the real eigenvalue of greatest real part. Let D be the real, block-diagonal canonicalform of A and Ae so that D = X−1AX = Y −1AeY where the real and imaginarycomponents of the (generalized) eigenvectors of A and Ae are the columns of X and Yrespectively (further, let the first eigenvector x1 > 0 be the first column of X). ThenAe = Y X−1AXY −1, but it is not necessarily true that XY −1 ≥ 0, as required.Let An = Ae + rI with r chosen large enough that all diagonal elements of An arepositive and An is nonsingular. Then An is nonnegative and primitive (irreducible, withonly one eigenvalue of maximum modulus; see, e.g., [11]). Then a further change ofbasis using (1/µ1An)k, where µ1 = λ1 + r is the first eigenvalue of An, yields Ae =(1/µ1An)−kY X−1AXY −1(1/µ1An)k. Since An is primitive and nonnegative, then ask → ∞, (1/µ1An)k → y1zT1 , where y1 is the first eigenvector of An (and of Ae), z1is the corresponding first left eigenvector, y1 > 0, z1 > 0, and yT1 z1 = 1. Hence,XY −1(1/µ1An)k → x1zT1 > 0. Therefore, for some finite k this change of basis B willbe nonnegative, Ae = B−1AB with B ≥ 0, and A is quasimonotone nondecreasing withrespect to a nonnegative cone. uunionsqWe note that the above theorem has, as a special case, our previous result for ma-trices with real spectra, since such shifted spectra always solve the nonnegative inverseeigenvalue problem, and we showed (using a different technique) that a positive firsteigenvector was a sufficient condition for quasimonotonicity with respect to a nonnega-tive cone. Furthermore, our previous result held only for n ≥ 3, as does this one as well,since a necessary condition for quasimonotonicity (a real eigenvalue) and the generalspectrum (implying a pair of complex eigenvalues) require at least three eigenvalues.References[1] P. Beaver and D. Canright, “The Quasimonotonicity of Linear Differential Systems”, Applic.Analysis, 70 (1–2) (1998), 67–73.[2] A. Berman, M. Neuman, and R.J. Stern, Nonnegative Matrices in Dynamical Systems,Wiley-Interscience, New York, 1991.[3] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systemand Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1994.[4] M. Boyle and D. Handleman, “The Spectra of Nonnegative Matrices via Symbolic Dynam-ics”, Annals of Mathematics, 133 (1991), 249–316.[5] M.T. Chu, “Inverse Eigenvalue Problems”, SIAM Review, 40 (1) (1998), 1–39.[6] S. Heikkila¨, On the Quasimonotonicity of Linear Differential Systems, Applic. Analysis, 10(1980), 121–126.4 Quasimon. Of Lin. Sys.—Complex Spectrum[7] S. Ko¨ksal and D.W. Fausett, “Projections, Cones, and Quasimonotonicity of Linear Differ-ential System”, Invited talk at the Second International Conference on Dynamic Systemsand Applications, Atlanta, GA, 24–27 May 1995.[8] V. Lakshmikantham and S. Leela, “Cone-Valued Lyapunov Functions”, J. Nonlinear Anal-ysis, 1 (1977), 215–222.[9] V. Lakshmikantham, V.M. Matrosov, and S. Sivasundaram, Vector Lyapunov Functions andStability Analysis of Nonlinear Systems, Kluwer Academic Publishers, The Netherlands,1991.[10] H. Schneider and V. Vidyasagar, “Cross-Positive Matrices”, SIAM J. Numer. Anal., 7 (4)(1970), 508–519.[11] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.

The Quasimonotonicity of Linear Differential Systems - The Complex Spectrum

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