Common diagonal solutions for second order complex family

Abstract

In this paper for second order complex interval matrix family we consider the problem of existence and evaluation of common diagonal solutions for the Lyapunov matrix inequality. Necessary and sufficient conditions are given. Number of examples are given, where known sufficient conditions do not give common solutions.

Keywords

• Complex Interval Matrices,Lyapunov Inequality,Diagonal Stability,Common Diagonal Solution

References

1. E.~Kaszkurewicz, A.~Bhaya. Matrix Diagonal Stability in Systems and Computation. Birkhauser, Boston, 2000.
2. O.~Pastravanu, M.-H.~Matcovschi. Diagonal stability of interval matrices and applications. Linear Algebra and its Applications, 433(8):1646--1658, 2010.
3. H.K.~Wimmer. Diagonal matrix solutions of a discrete-time Lyapunov inequality. IEEE Transactions on Automatic Control, 43(3): 442--445, 1998.
4. C.R.~Johnson, T.D.~Lenker, S.K.~Narayan. Schur-type stability properties and complex diagonal scaling. Linear Algebra and its Applications , 429(10):2497--2520, 2008.
5. V.~Dzhafarov, T.~Büyükköroğlu. On the stability of a convex set of matrices. Linear Algebra and its Applications, 414(2): 547--559, 2006.
6. B.N.~Datta. Stability and inertia. Linear Algebra and its Applications, 302:563--600,1999.
7. G.W.~Cross. Three types of matrix stability. Linear Algebra and its Applications , 20(3):253--263, 1978.
8. G.~Barker, A.~Berman, R.~Plemmons. Positive diagonal solutions to the Lyapunov equations. Linear and Multilinear Algebra, 5(4): 249--256, 1978.

9. H.K.~Khalil. On the existence of positive diagonal $P$ such that $\uppercase{PA + A^T P} < 0$. IEEE Transactions on Automatic Control , 27(1): 181--184, 1982.
10. A.~Berman, D.~Hershkowitz. Matrix diagonal stability and its implications. SIAM Journal on Algebraic Discrete Methods, 4(3):377--382,1983.
11. D.~Hershkowitz, H.~Schneider. Lyapunov diagonal semistability of real $H$-matrices. Linear Algebra and its Applications, 71:119--149, 1985.
12. M.~Arcak, E.D.~Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica , 42(9):1531--1537, 2006.
13. O.~Mason, R.~Shorten. On the simultaneous diagonal stability of a pair of positive linear systems. Linear Algebra and its Applications , 413(1):13--23, 2006.
14. B.~Yıldız T.~Büyükkoroğlu and V.~Dzhafarov. Common diagonal solutions to the Lyapunov inequalities for interval systems. Journal of Inequalities and Applications, 255:1--12, 2016.
15. O.~Pastravanu, M.-H.~Matcovschi. Sufficient conditions for Schur and Hurwitz diagonal stability of complex interval matrices. Linear Algebra and its Applications, 467:149--173, 2015.
16. T.~Büyükkoroğlu,Common diagonal Lyapunov function for third order linear switched system. Journal of Computational and Applied Mathematics, 236(15):3647--3653, 2012.}