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doi: 10.3934/eect.2022008
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Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle

1. 

Department of Mathematics, Eastern Mediterranean University, Mersin 10, 99628, T.R. North Cyprus, Turkey

2. 

Departamento de Estatística, Instituto de Matematicas, University of Santiago de Compostela, 15782, Spain

*Corresponding author: Nazim I. Mahmudov

Received  September 2021 Revised  December 2021 Early access February 2022

In this paper, we obtain a closed-form representation of a mild solution to the fractional stochastic degenerate evolution equation in a Hilbert space using the subordination principle and semigroup theory. We study aforesaid abstract fractional stochastic Cauchy problem with nonlinear state-dependent terms and show that if the Sobolev type resolvent families describing the linear part of the model are exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions for nonlinearity. We also establish conditions for stabilizability and prove that the stochastic nonlinear fractional Cauchy problem is exponentially stabilizable when the stabilizer acts linearly on the control systems. Finally, we provide applications to show the validity of our theory.

Citation: Arzu Ahmadova, Nazim I. Mahmudov, Juan J. Nieto. Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle. Evolution Equations and Control Theory, doi: 10.3934/eect.2022008
References:
[1]

L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces, 2015 (2015), Art. ID 158145, 9 pp. doi: 10.1155/2015/158145.

[2]

R. AgarwalD. BaleanuJ. J. NietoD. Torre and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[3]

A. Ahmadova, I. T. Huseynov, A. Fernandez and N. I. Mahmudov, Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 97 (2021), Paper No. 105735, 23 pp. doi: 10.1016/j.cnsns.2021.105735.

[4]

A. Ahmadova and N. I. Mahmudov, Existence and uniqueness results for a class of fractional stochastic neutral differential equations, Chaos Solitons Fractals, 139 (2020), 110253, 8 pp. doi: 10.1016/j.chaos.2020.110253.

[5]

M. Ait Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic control, IEEE Trans. Autom. Control., 45 (2000), 1131-1143.  doi: 10.1109/9.863597.

[6]

K. BalachandranS. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comp. Math. Appl., 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.

[7]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.

[8]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-Heidelberg-New York, 1978.

[9]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
[11]

H. DengM. Kristić and R. William, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE Trans. Automat. Control, 46 (2001), 1237-1253.  doi: 10.1109/9.940927.

[12]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.

[13]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 215. Marcel Dekker, Inc., New York, 1999.

[14]

M. FeckanJ. Wang and Y. Zhou, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators, J. Optim. Theory Appl., 156 (2013), 79-95.  doi: 10.1007/s10957-012-0174-7.

[15]

R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-662-43930-2.

[16]

W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Academic Verlag, Berlin, 1995.

[17]

I. T. Huseynov, A. Ahmadova, A. Fernandez and N. I. Mahmudov, Explicit analytic solutions of incommensurate fractional differential equation systems, Appl. Math. Comput., 390 (2021), Paper No. 125590, 21 pp. doi: 10.1016/j.amc.2020.125590.

[18]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Springer, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.

[19]

V. Keyantuo, C. Lizama and M. Warma, Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal., 2013 (2013), Art. ID 614328, 11 pp. doi: 10.1155/2013/614328.

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Sceince B. V., 2006.

[21]

H.-W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie, AkademieVerlag, Berlin, 1986.

[22] H. J. Kushner, Stochastic Stability and Control, Academic Press, New York, 1967. 
[23]

I. Lasiecka and R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlinear Anal., 121 (2015), 424-446.  doi: 10.1016/j.na.2015.03.012.

[24]

F. LiJ. Liang and H. K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 391 (2012), 510-525. 

[25]

G. Li and M. Chen, Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems, Adv. Difference Equ., 2015 (2015), 14, 12 pp. doi: 10.1186/s13662-014-0342-1.

[26]

J. Liang and T. J. Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J. Math. Anal. Appl., 259 (2001), 398-412.  doi: 10.1006/jmaa.2000.7406.

[27]

X. LiuY. Li and W. Zhang, Stochastic linear quadratic optimal control with constraint for discrete-time systems, Appl. Math. Comput., 228 (2014), 264-270.  doi: 10.1016/j.amc.2013.09.036.

[28]

X. Lin and R. Zhang, $H_{\infty}$ control for stochastic systems with poisson jumps, J. Syst. Sci. Complex., 24 (2011), 683-700.  doi: 10.1007/s11424-011-9085-1.

[29]

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag London Ltd., 1999. doi: 10.1007/978-1-4471-0419-3.

[30]

N. I. Mahmudov, Existence and approximate controllability of Sobolev type fractional stochastic evolution equations, Bull. Pol. Acad. Sci., 62 (2014), 205-215.  doi: 10.2478/bpasts-2014-0020.

[31]

N. I. Mahmudov, Necessary first-order and second-order optimality conditions in discrete-time stochastic systems, J. Optim. Theory Appl., 182 (2019), 1001-1018.  doi: 10.1007/s10957-019-01478-y.

[32]

N. I. Mahmudov, A. Ahmadova and I. T. Huseynov, A new technique for solving Sobolev type fractional multi-order evolution equations, arXiv: 2102.10318.

[33]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.

[34]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastics Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102.

[35]

M. G. Mittag-Leffler, Sopra la funzione $E_{\alpha}(x)$, Rend. R. Acc. Lincei., 13 (1904), 3-5. 

[36]

P. H. A. Ngoc, A new approach to mean square exponential stability of stochastic functional differential equations, IEEE Control Syst. Lett., 5 (2021), 1645-1650.  doi: 10.1109/LCSYS.2020.3042479.

[37]

S. Nicaise and C. Pignotti, Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.  doi: 10.1007/s00028-018-0427-5.

[38]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 5$^{th}$ edition, Springer-Verlag, Heidelberg, 1998. doi: 10.1007/978-3-662-03620-4.

[39] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[40]

R. Ponce, Subordination principle for fractional diffusion wave equations of Sobolev type, Fract. Calc. Appl. Anal., 23 (2020), 427-449.  doi: 10.1515/fca-2020-0021.

[41]

A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite dimensional systems, SIAM Rev., 23 (1981), 25-52.  doi: 10.1137/1023003.

[42]

R. Rebarber and H. Zwart, Open-loop stabilizability of infinite-dimensional systems, Math. Control Signals Systems, 11 (1998), 129-160.  doi: 10.1007/BF02741888.

[43]

P. RevathiR. Sakthivel and Y. Ren, Stochastic functional differential equations of Sobolev-type with infinite delay, Stat. Probab. Lett., 109 (2016), 68-77.  doi: 10.1016/j.spl.2015.10.019.

[44]

R. K. SaxenaS. L. Kalla and R. Saxena, Multivariate analogue of generalised Mittag-Leffler function, Integral Transforms Spec. Funct., 22 (2011), 533-548.  doi: 10.1080/10652469.2010.533474.

[45]

K. ShahA. Ullah and J. J. Nieto, Study of fractional order impulsive evolution problem under nonlocal Cauchy conditions, Math. Meth. Appl. Sci., 44 (2021), 8516-8527.  doi: 10.1002/mma.7274.

[46]

K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3712-6.

[47]

E. D. Sontag, Mathematical Control Theory, volume 6 of Texts in Applied Mathematics, Springer-Verlag, New York, second edition, 1998. doi: 10.1007/978-1-4612-0577-7.

[48]

R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl., 52 (1975), 382-403.  doi: 10.1016/0022-247X(75)90067-0.

[49]

J. WangM. Feckan and Y. Zhou, Controllability of Sobolev type fractional evolution systems, Dyn. Partial Differ. Equ., 11 (2014), 71-87.  doi: 10.4310/DPDE.2014.v11.n1.a4.

[50]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[51]

W. H. Zhang and B. S. Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica, 40 (2004), 87-94.  doi: 10.1016/j.automatica.2003.07.002.

[52]

H. ZitaneA. Boutoulout and F. M. Torres Delfim, The stability and stabilization of infinite dimensional caputo-time fractional differential linear systems, Mathematics, 8 (2020), 353.  doi: 10.3390/math8030353.

show all references

References:
[1]

L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces, 2015 (2015), Art. ID 158145, 9 pp. doi: 10.1155/2015/158145.

[2]

R. AgarwalD. BaleanuJ. J. NietoD. Torre and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[3]

A. Ahmadova, I. T. Huseynov, A. Fernandez and N. I. Mahmudov, Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 97 (2021), Paper No. 105735, 23 pp. doi: 10.1016/j.cnsns.2021.105735.

[4]

A. Ahmadova and N. I. Mahmudov, Existence and uniqueness results for a class of fractional stochastic neutral differential equations, Chaos Solitons Fractals, 139 (2020), 110253, 8 pp. doi: 10.1016/j.chaos.2020.110253.

[5]

M. Ait Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic control, IEEE Trans. Autom. Control., 45 (2000), 1131-1143.  doi: 10.1109/9.863597.

[6]

K. BalachandranS. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comp. Math. Appl., 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.

[7]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.

[8]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-Heidelberg-New York, 1978.

[9]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
[11]

H. DengM. Kristić and R. William, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE Trans. Automat. Control, 46 (2001), 1237-1253.  doi: 10.1109/9.940927.

[12]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.

[13]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 215. Marcel Dekker, Inc., New York, 1999.

[14]

M. FeckanJ. Wang and Y. Zhou, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators, J. Optim. Theory Appl., 156 (2013), 79-95.  doi: 10.1007/s10957-012-0174-7.

[15]

R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-662-43930-2.

[16]

W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Academic Verlag, Berlin, 1995.

[17]

I. T. Huseynov, A. Ahmadova, A. Fernandez and N. I. Mahmudov, Explicit analytic solutions of incommensurate fractional differential equation systems, Appl. Math. Comput., 390 (2021), Paper No. 125590, 21 pp. doi: 10.1016/j.amc.2020.125590.

[18]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Springer, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.

[19]

V. Keyantuo, C. Lizama and M. Warma, Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal., 2013 (2013), Art. ID 614328, 11 pp. doi: 10.1155/2013/614328.

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Sceince B. V., 2006.

[21]

H.-W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie, AkademieVerlag, Berlin, 1986.

[22] H. J. Kushner, Stochastic Stability and Control, Academic Press, New York, 1967. 
[23]

I. Lasiecka and R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlinear Anal., 121 (2015), 424-446.  doi: 10.1016/j.na.2015.03.012.

[24]

F. LiJ. Liang and H. K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 391 (2012), 510-525. 

[25]

G. Li and M. Chen, Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems, Adv. Difference Equ., 2015 (2015), 14, 12 pp. doi: 10.1186/s13662-014-0342-1.

[26]

J. Liang and T. J. Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J. Math. Anal. Appl., 259 (2001), 398-412.  doi: 10.1006/jmaa.2000.7406.

[27]

X. LiuY. Li and W. Zhang, Stochastic linear quadratic optimal control with constraint for discrete-time systems, Appl. Math. Comput., 228 (2014), 264-270.  doi: 10.1016/j.amc.2013.09.036.

[28]

X. Lin and R. Zhang, $H_{\infty}$ control for stochastic systems with poisson jumps, J. Syst. Sci. Complex., 24 (2011), 683-700.  doi: 10.1007/s11424-011-9085-1.

[29]

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag London Ltd., 1999. doi: 10.1007/978-1-4471-0419-3.

[30]

N. I. Mahmudov, Existence and approximate controllability of Sobolev type fractional stochastic evolution equations, Bull. Pol. Acad. Sci., 62 (2014), 205-215.  doi: 10.2478/bpasts-2014-0020.

[31]

N. I. Mahmudov, Necessary first-order and second-order optimality conditions in discrete-time stochastic systems, J. Optim. Theory Appl., 182 (2019), 1001-1018.  doi: 10.1007/s10957-019-01478-y.

[32]

N. I. Mahmudov, A. Ahmadova and I. T. Huseynov, A new technique for solving Sobolev type fractional multi-order evolution equations, arXiv: 2102.10318.

[33]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.

[34]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastics Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102.

[35]

M. G. Mittag-Leffler, Sopra la funzione $E_{\alpha}(x)$, Rend. R. Acc. Lincei., 13 (1904), 3-5. 

[36]

P. H. A. Ngoc, A new approach to mean square exponential stability of stochastic functional differential equations, IEEE Control Syst. Lett., 5 (2021), 1645-1650.  doi: 10.1109/LCSYS.2020.3042479.

[37]

S. Nicaise and C. Pignotti, Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.  doi: 10.1007/s00028-018-0427-5.

[38]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 5$^{th}$ edition, Springer-Verlag, Heidelberg, 1998. doi: 10.1007/978-3-662-03620-4.

[39] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[40]

R. Ponce, Subordination principle for fractional diffusion wave equations of Sobolev type, Fract. Calc. Appl. Anal., 23 (2020), 427-449.  doi: 10.1515/fca-2020-0021.

[41]

A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite dimensional systems, SIAM Rev., 23 (1981), 25-52.  doi: 10.1137/1023003.

[42]

R. Rebarber and H. Zwart, Open-loop stabilizability of infinite-dimensional systems, Math. Control Signals Systems, 11 (1998), 129-160.  doi: 10.1007/BF02741888.

[43]

P. RevathiR. Sakthivel and Y. Ren, Stochastic functional differential equations of Sobolev-type with infinite delay, Stat. Probab. Lett., 109 (2016), 68-77.  doi: 10.1016/j.spl.2015.10.019.

[44]

R. K. SaxenaS. L. Kalla and R. Saxena, Multivariate analogue of generalised Mittag-Leffler function, Integral Transforms Spec. Funct., 22 (2011), 533-548.  doi: 10.1080/10652469.2010.533474.

[45]

K. ShahA. Ullah and J. J. Nieto, Study of fractional order impulsive evolution problem under nonlocal Cauchy conditions, Math. Meth. Appl. Sci., 44 (2021), 8516-8527.  doi: 10.1002/mma.7274.

[46]

K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3712-6.

[47]

E. D. Sontag, Mathematical Control Theory, volume 6 of Texts in Applied Mathematics, Springer-Verlag, New York, second edition, 1998. doi: 10.1007/978-1-4612-0577-7.

[48]

R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl., 52 (1975), 382-403.  doi: 10.1016/0022-247X(75)90067-0.

[49]

J. WangM. Feckan and Y. Zhou, Controllability of Sobolev type fractional evolution systems, Dyn. Partial Differ. Equ., 11 (2014), 71-87.  doi: 10.4310/DPDE.2014.v11.n1.a4.

[50]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[51]

W. H. Zhang and B. S. Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica, 40 (2004), 87-94.  doi: 10.1016/j.automatica.2003.07.002.

[52]

H. ZitaneA. Boutoulout and F. M. Torres Delfim, The stability and stabilization of infinite dimensional caputo-time fractional differential linear systems, Mathematics, 8 (2020), 353.  doi: 10.3390/math8030353.

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