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Lyapunov inequalities for discrete linear Hamiltonian systems. (English) Zbl 1055.39029
The authors study the discrete system $$\aligned x(t+1)-x(t)&=a(t)x(t+1)+b(t)u(t)\\ u(t+1)-u(t)&=-c(t)x(t+1)-a(t)u(t). \endaligned\tag1$$ They derive some Lyapunov type inequalities from (1). They use these inequalities to derive a disconjugacy criterion. Then they study the stability of (1) when the coefficients $a(t)$, $b(t)$ and $c(t)$ are periodic. They derive sufficient conditions for the instability and stability of (1). A remark is also given for the stability of the corresponding continuous time system with periodic coefficients.

39A12Discrete version of topics in analysis
39A11Stability of difference equations (MSC2000)
93C55Discrete-time control systems
93D05Lyapunov and other classical stabilities of control systems
37J25Stability problems (finite-dimensional Hamiltonian etc. systems)
34D20Stability of ODE
Full Text: DOI
[1] Krein, M. G.: Amer. math. Soc. translations, ser. 2. 120, 1-70 (1983)
[2] Yakubovich, V. A.; Starzhinsky, V. M.: Linear differential equations with periodic coefficients, part I and II. (1975)
[3] Cheng, S. S.: Lyapunov inequalities for differential and difference equations. Fasc. math. 23, 25-41 (1991) · Zbl 0753.34017
[4] Agarwal, R.; Ahlbrandt, C. D.; Bohner, M.; Peterson, A. C.: Discrete linear Hamiltonian systems: A survey. Dynam. systems appl. 8, 307-333 (1999) · Zbl 0942.39009
[5] Ahlbrandt, C. D.; Peterson, A. C.: Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations. (1996) · Zbl 0860.39001
[6] Erbe, L.; Yan, P.: Disconjugacy for linear Hamiltonian difference systems. J. math. Anal. appl. 167, 355-367 (1992) · Zbl 0762.39003
[7] Hartman, P.: Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. amer. Math. soc. 246, 1-30 (1978) · Zbl 0409.39001
[8] Atici, F. M.; Guseinov, G. Sh.: Criteria for the stability of second order difference equations with periodic coefficients. Communications in appl. Anal. 3, 503-515 (1999) · Zbl 0933.39013
[9] Atici, F. M.; Guseinov, G. Sh.; Kaymakçalan, B.: On Lyapunov inequality in stability theory for Hill’s equation on time scales. J. inequal. Appl. 5, 603-620 (2000) · Zbl 0971.39005
[10] Bohner, M.: Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. J. math. Anal. appl. 199, 804-826 (1996) · Zbl 0855.39018
[11] Bohner, M.; Clark, S.; Ridenhour, J.: Lyapunov inequalities for time scales. J. inequal. Appl. 7, 61-77 (2002) · Zbl 1088.34503
[12] Clark, S.; Hinton, D. B.: A Liapunov inequality for linear Hamiltonian systems. Math. inequal. Appl. 1, 201-209 (1998) · Zbl 0909.34033
[13] Clark, S.; Hinton, D. B.: Discrete Lyapunov inequalities. Dynam. systems appl. 8, 369-380 (1999) · Zbl 0940.39013
[14] Guseinov, G. Sh.; Kaymakçalan, B.: On a disconjugacy criterion for second order dynamic equations on time scales. J. comput. Appl. math. 141, 187-196 (2002) · Zbl 1014.34023
[15] Halanay, A.; Rasvan, V.: Stability and boundary value problems for discrete-time linear Hamiltonian systems. Dynam. systems appl. 8, 439-459 (1999)
[16] Elaydi, S. N.: An introduction to difference equations. (1996) · Zbl 0840.39002