# American Institute of Mathematical Sciences

October  2017, 22(8): 3167-3197. doi: 10.3934/dcdsb.2017169

## Exponential stability for nonautonomous functional differential equations with state-dependent delay

Received  January 2016 Revised  January 2017 Published  June 2017

Fund Project: Partly supported by MEC (Spain) under project MTM2015-66330-P and by European Commission under project H2020-MSCA-ITN-2014.

The properties of stability of a compact set $\mathcal{K}$ which is positively invariant for a semiflow $(\Omega× {W^{1,\infty }}([-r,0],\mathbb{R}^n),Π,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $\mathcal{K}$ induce linear skew-product semiflows on the bundles $\mathcal{K}×{W^{1,\infty }}([-r,0],\mathbb{R}^n)$ and $\mathcal{K}× C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $\mathcal{K}$ in $\Omega×{W^{1,\infty }}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this compact set when the supremum norm is taken in ${W^{1,\infty }}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.

Citation: Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
##### References:
 [1] O. Arino, E. Sánchez and A. Fathallah, State-dependent delay differential equations in populations dynamics: Modeling and analysis, Fields Inst. Commun., 29 (2001), 19-36. [2] M. V. Barbarossa and H. O. Walther, Linearized stability for a new class of neutral equations with state-dependent delay, Differ. Equ. Dyn. Syst., 24 (2016), 63-79.  doi: 10.1007/s12591-014-0204-z. [3] Y. Chen, Q. Hu and J. Wu, Second-order differentiability with respect to parameters for differential equations with adaptative delays, Front. Math. China, 5 (2010), 221-286.  doi: 10.1007/s11464-010-0005-9. [4] S.-N. Chow and H. Leiva, Dynamical spectrum for time dependent linear systems in Banach spaces, Japan J. Indust. Appl. Math., 11 (1994), 379-415.  doi: 10.1007/BF03167229. [5] S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations, 120 (1995), 429-477.  doi: 10.1006/jdeq.1995.1117. [6] J. K. Hale and S. M. Verdyun Lunel, Introduction to Functional Differential Equations Appl. Math. Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [7] F. Hartung, On differentiability of solutions with respect to parameters in a class of functional differential equations, Funct. Differ. Equ., 4 (1997), 65-79. [8] F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Appl. Math., 174 (2005), 201-211.  doi: 10.1016/j.cam.2004.04.006. [9] F. Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differential Equations, 23 (2011), 843-884.  doi: 10.1007/s10884-011-9218-1. [10] F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, J. Dynam. Differential Equations, 25 (2013), 1089-1138.  doi: 10.1007/s10884-013-9330-5. [11] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, Elsevier, North-Holland, 3 (2006), 435-545.  doi: 10.1016/S1874-5725(06)80009-X. [12] X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅰ: Finitely differentiable, hyperbolic case, J. Dynam. Differential Equations (2016). [13] X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅱ: Analytic case, J. Differential Equations, 261 (2016), 2068-2108.  doi: 10.1016/j.jde.2016.04.024. [14] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Math. , 1473, Springer-Verlag, Berlin, Heidelberg, New York, 1991. doi: 10.1007/BFb0084432. [15] Q. Hu, J. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations, SIAM J. Math. Anal., 44 (2012), 2401-2427.  doi: 10.1137/100793712. [16] Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), 2801-2840.  doi: 10.1016/j.jde.2010.03.020. [17] T. Insperger and G. Stépán, Semi-discretation for Time-Delay Systems. Stability and Engeneering Applications, Appl. Math. Sci. , 178 Springter, New York, 2011. [18] T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold, J. Differential Equations, 260 (2016), 4454-4472.  doi: 10.1016/j.jde.2015.11.018. [19] J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4085-4103.  doi: 10.1016/j.jde.2010.10.023. [20] I. Maroto, C. Núñez and R. Obaya, Dynamical properties of nonautonomous functional differential equations with state-dependent delay, Discrete Cont. Dynam. Syst., Ser. A, 37 (2017), 3939-3961.  doi: 10.3934/dcds.2017167. [21] S. Novo, R. Obaya and A. M. Sanz, Exponential stability in non-autonomous delayed equations with applications to neural networks, Discrete Contin. Dyn. Syst., 18 (2007), 517-536.  doi: 10.3934/dcds.2007.18.517. [22] S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646.  doi: 10.1016/j.jde.2006.12.009. [23] R. J. Sacker and G. R. Sell, Lifting properties in skew-products flows with applications to differential equations Mem. Amer. Math. Soc. Amer. Math. Soc. , Providence, 11 (1977), iv+67 pp. doi: 10.1090/memo/0190. [24] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67.  doi: 10.1006/jdeq.1994.1113. [25] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows Mem. Amer. Math. Soc. , 136 (1998), x+93 pp. doi: 10.1090/memo/0647. [26] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Science Appl. Math. , 57, Springer, New York, 2001. doi: 10.1007/978-1-4419-7646-8. [27] H. O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65.  doi: 10.1016/j.jde.2003.07.001. [28] H. O. Walther, Smoothness of semiflows for differential equations with state-dependent delays, J. Math. Sci. 124 (2004). [29] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Nonlinear Analysis and Aplications, 6, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110879971.

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##### References:
 [1] O. Arino, E. Sánchez and A. Fathallah, State-dependent delay differential equations in populations dynamics: Modeling and analysis, Fields Inst. Commun., 29 (2001), 19-36. [2] M. V. Barbarossa and H. O. Walther, Linearized stability for a new class of neutral equations with state-dependent delay, Differ. Equ. Dyn. Syst., 24 (2016), 63-79.  doi: 10.1007/s12591-014-0204-z. [3] Y. Chen, Q. Hu and J. Wu, Second-order differentiability with respect to parameters for differential equations with adaptative delays, Front. Math. China, 5 (2010), 221-286.  doi: 10.1007/s11464-010-0005-9. [4] S.-N. Chow and H. Leiva, Dynamical spectrum for time dependent linear systems in Banach spaces, Japan J. Indust. Appl. Math., 11 (1994), 379-415.  doi: 10.1007/BF03167229. [5] S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations, 120 (1995), 429-477.  doi: 10.1006/jdeq.1995.1117. [6] J. K. Hale and S. M. Verdyun Lunel, Introduction to Functional Differential Equations Appl. Math. Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [7] F. Hartung, On differentiability of solutions with respect to parameters in a class of functional differential equations, Funct. Differ. Equ., 4 (1997), 65-79. [8] F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Appl. Math., 174 (2005), 201-211.  doi: 10.1016/j.cam.2004.04.006. [9] F. Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differential Equations, 23 (2011), 843-884.  doi: 10.1007/s10884-011-9218-1. [10] F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, J. Dynam. Differential Equations, 25 (2013), 1089-1138.  doi: 10.1007/s10884-013-9330-5. [11] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, Elsevier, North-Holland, 3 (2006), 435-545.  doi: 10.1016/S1874-5725(06)80009-X. [12] X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅰ: Finitely differentiable, hyperbolic case, J. Dynam. Differential Equations (2016). [13] X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅱ: Analytic case, J. Differential Equations, 261 (2016), 2068-2108.  doi: 10.1016/j.jde.2016.04.024. [14] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Math. , 1473, Springer-Verlag, Berlin, Heidelberg, New York, 1991. doi: 10.1007/BFb0084432. [15] Q. Hu, J. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations, SIAM J. Math. Anal., 44 (2012), 2401-2427.  doi: 10.1137/100793712. [16] Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), 2801-2840.  doi: 10.1016/j.jde.2010.03.020. [17] T. Insperger and G. Stépán, Semi-discretation for Time-Delay Systems. Stability and Engeneering Applications, Appl. Math. Sci. , 178 Springter, New York, 2011. [18] T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold, J. Differential Equations, 260 (2016), 4454-4472.  doi: 10.1016/j.jde.2015.11.018. [19] J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4085-4103.  doi: 10.1016/j.jde.2010.10.023. [20] I. Maroto, C. Núñez and R. Obaya, Dynamical properties of nonautonomous functional differential equations with state-dependent delay, Discrete Cont. Dynam. Syst., Ser. A, 37 (2017), 3939-3961.  doi: 10.3934/dcds.2017167. [21] S. Novo, R. Obaya and A. M. Sanz, Exponential stability in non-autonomous delayed equations with applications to neural networks, Discrete Contin. Dyn. Syst., 18 (2007), 517-536.  doi: 10.3934/dcds.2007.18.517. [22] S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations, 235 (2007), 623-646.  doi: 10.1016/j.jde.2006.12.009. [23] R. J. Sacker and G. R. Sell, Lifting properties in skew-products flows with applications to differential equations Mem. Amer. Math. Soc. Amer. Math. Soc. , Providence, 11 (1977), iv+67 pp. doi: 10.1090/memo/0190. [24] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67.  doi: 10.1006/jdeq.1994.1113. [25] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows Mem. Amer. Math. Soc. , 136 (1998), x+93 pp. doi: 10.1090/memo/0647. [26] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Science Appl. Math. , 57, Springer, New York, 2001. doi: 10.1007/978-1-4419-7646-8. [27] H. O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65.  doi: 10.1016/j.jde.2003.07.001. [28] H. O. Walther, Smoothness of semiflows for differential equations with state-dependent delays, J. Math. Sci. 124 (2004). [29] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Nonlinear Analysis and Aplications, 6, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110879971.
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