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doi: 10.3934/eect.2022014
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On analytic semigroup generators involving Caputo fractional derivative

Institute of Control Engineering and Robotics, The AGH University of Science and Technology, Kraków, Poland

Received  August 2021 Revised  February 2022 Early access March 2022

Our investigations are motivated by the well - posedness problem of some dynamical models with anomalous diffusion described by the Caputo spatial fractional derivative of order $ \alpha \in (1, 2) $. We propose a characterization of an exponentially stable analytic semigroup generator using the inverse operator. This characterization enables us to establish the form of a generator involving the Caputo fractional derivative, under various boundary conditions. In particular, the results simplify those known from literature obtained by means of the fractional Sobolev spaces and some perturbation results. Going further, we show how to construct a control system in factor form, having such a generator as the state operator.

Citation: Piotr Grabowski. On analytic semigroup generators involving Caputo fractional derivative. Evolution Equations and Control Theory, doi: 10.3934/eect.2022014
References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector–Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

B. BaeumerM. Kovács and H. Sankaranarayanan, Fractional partial differential equations with boundary conditions, J. Differential Equations, 264 (2018), 1377-1410.  doi: 10.1016/j.jde.2017.09.040.

[3]

A. V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics, No. 3. Springer-Verlag, New York-Heidelberg, 1976.

[4]

H. Bateman, A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. vol. I and II, New York: McGraw–Hill, 1954.

[5]

R. DeLaubenfels, Inverses of generators, Proc. Amer. Math. Soc., 104 (1988), 443-448.  doi: 10.1090/S0002-9939-1988-0962810-6.

[6]

M. M. Dzhrbashian, Integral Transforms and Representation of Functions in the Complex Domain, Moscow: Nauka., 1966 (in Russian).

[7]

M. M. Dzhrbashian and A. B. Nersesian, Expansion in special biorthogonal systems and boundary value problems for fractional order differential equations, Soviet Math. Dokl, 1 (1960), 629-633. 

[8]

A. M. A. El–Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, 3 (2006), 81-95. 

[9]

K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext. Springer, New York, 2006.

[10]

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

[11]

R. GorenfloY. Luchko and M. Yamamoto, Time–fractional diffusion equations in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048.

[12]

P. Grabowski, On spectral–Lyapunov approach to parametric optimization of distributed parameter systems, IMA J. Math. Control Inform., 7 (1990), 317-338.  doi: 10.1093/imamci/7.4.317.

[13]

P. Grabowski, Some modifications of the Weiss–Staffans perturbation theorem, Internat. J. Robust Nonlinear Control, 27 (2017), 1094-1121.  doi: 10.1002/rnc.3617.

[14]

P. Grabowski, Small-gain theorem for a class of abstract parabolic systems, Opuscula Math., 38 (2018), 651-680.  doi: 10.7494/OpMath.2018.38.5.651.

[15]

G. M. Gubreev, The basic property of families of Mittag–Leffler type functions, Soviet J. Contemporary Math. Anal., 23 (1988), 43-78. 

[16]

G. M. Gubreev, Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights, J. Math. Sci., 71 (1994), 2192-2221.  doi: 10.1007/BF02111293.

[17]

P. R. Halmos and V. S. Sunder, Bounded Integral Operators on $\text{L}^{2}$ Spaces, Berlin: Springer. 1978.

[18]

J. W. Hanneken, B. N. Narahari Achar and D. M. Vaught, An alpha–beta phase diagram representation of the zeros and properties of the Mittag–Leffler function, Adv. Math. Phys., (2013), Art. ID 421685, 13 pp. doi: 10.1155/2013/421685.

[19]

K. ItoB. Jin and T. Takeuchi, On the sectorial property of the Caputo derivative operator, Appl. Math. Lett., 47 (2015), 43-46.  doi: 10.1016/j.aml.2015.03.001.

[20]

B. JinR. LazarovJ. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.  doi: 10.1090/mcom/2960.

[21]

B. Jin and W. Rundell, An inverse Sturm-Liouville problem with a fractional derivative, J. Comput. Phys., 231 (2012), 4954-4966.  doi: 10.1016/j.jcp.2012.04.005.

[22]

A. P. Khromov, Finite–dimensional perturbations of Volterra operators, J. Math. Sci. (N.Y.), 138 (2006), 5893-6066.  doi: 10.1007/s10958-006-0346-9.

[23]

A. M. Nakhushev, The Sturm-Liouville problem for an ordinary second–order differential equation with fractional derivatives in the lowest terms, Soviet Mathematics, Doklady, 18 (1977), 666-670. 

[24]

T. Namba, P. Rybka and V. R. Voller, Some comments on using fractional derivative operators in modeling non-local diffusion processes, J. Comput. Appl. Math., 381 (2021), Paper No. 113040, 17 pp. doi: 10.1016/j.cam.2020.113040.

[25]

K. Oprzȩdkiewicz, Non integer order, state space model of heat transfer process using Caputo–Fabrizio operator, Bulletin of the Polish Academy of Sciences, Technical Sciences, 66 (2018), 249-255. 

[26]

K. Oprzȩdkiewicz and K. Dziedzic, New parameter identification method for the fractional order, state space model of heat transfer process, Automation, (2018), AISC 743,401–417.

[27]

K. Oprzȩdkiewicz and W. Mitkowski, A memory–efficient noninteger-order discrete time state-space model of a heat transfer process, Int. J. Appl. Math. Comput. Sci., 28 (2018), 49-659.  doi: 10.2478/amcs-2018-0050.

[28]

K. OprzȩdkiewiczW. MitkowskiK. Gawin and K. Dziedzic, The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process, Bulletin of the Polish Academy of Sciences, Technical Sciences, 66 (2018), 501-507. 

[29]

R. T. Parovik, An algorithm for computing the Mittag–Leffler function in the Maple symbolic mathematical package, International Journal of Soft Computing, 11 (2016), 487-491. 

[30]

M. Riesz, Intégrales de Riemann–Liouville et potentiels, Acta Universitatis Szegediensis, 9 (1938), 1–42,116–118.

[31]

K. Ryszewska, An analytic semigroup generated by a fractional differential operator, J. Math. Anal. Appl., 483 (2020), 123654, 17 pp. doi: 10.1016/j.jmaa.2019.123654.

[32]

K. Schmüdgen, Unbounded Self–adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, 265. Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.

[33]

S. TatarR. Tinaztepe and S. Ulusoy, Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal., 95 (2016), 1-23.  doi: 10.1080/00036811.2014.984291.

[34]

J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, 68. Springer-Verlag, New York-Berlin, 1980.

[35]

Q. Yang, Novel analytic and numerical methods for solving fractional dynamical systems, Ph.D. Thesis, School of Mathematical Sciences, Queensland University of Technology, (2010), 1–201, https://eprints.qut.edu.au/35750/1/Qianqian_Yang_Thesis.pdf.

show all references

References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector–Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

B. BaeumerM. Kovács and H. Sankaranarayanan, Fractional partial differential equations with boundary conditions, J. Differential Equations, 264 (2018), 1377-1410.  doi: 10.1016/j.jde.2017.09.040.

[3]

A. V. Balakrishnan, Applied Functional Analysis, Applications of Mathematics, No. 3. Springer-Verlag, New York-Heidelberg, 1976.

[4]

H. Bateman, A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. vol. I and II, New York: McGraw–Hill, 1954.

[5]

R. DeLaubenfels, Inverses of generators, Proc. Amer. Math. Soc., 104 (1988), 443-448.  doi: 10.1090/S0002-9939-1988-0962810-6.

[6]

M. M. Dzhrbashian, Integral Transforms and Representation of Functions in the Complex Domain, Moscow: Nauka., 1966 (in Russian).

[7]

M. M. Dzhrbashian and A. B. Nersesian, Expansion in special biorthogonal systems and boundary value problems for fractional order differential equations, Soviet Math. Dokl, 1 (1960), 629-633. 

[8]

A. M. A. El–Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, 3 (2006), 81-95. 

[9]

K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext. Springer, New York, 2006.

[10]

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

[11]

R. GorenfloY. Luchko and M. Yamamoto, Time–fractional diffusion equations in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048.

[12]

P. Grabowski, On spectral–Lyapunov approach to parametric optimization of distributed parameter systems, IMA J. Math. Control Inform., 7 (1990), 317-338.  doi: 10.1093/imamci/7.4.317.

[13]

P. Grabowski, Some modifications of the Weiss–Staffans perturbation theorem, Internat. J. Robust Nonlinear Control, 27 (2017), 1094-1121.  doi: 10.1002/rnc.3617.

[14]

P. Grabowski, Small-gain theorem for a class of abstract parabolic systems, Opuscula Math., 38 (2018), 651-680.  doi: 10.7494/OpMath.2018.38.5.651.

[15]

G. M. Gubreev, The basic property of families of Mittag–Leffler type functions, Soviet J. Contemporary Math. Anal., 23 (1988), 43-78. 

[16]

G. M. Gubreev, Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights, J. Math. Sci., 71 (1994), 2192-2221.  doi: 10.1007/BF02111293.

[17]

P. R. Halmos and V. S. Sunder, Bounded Integral Operators on $\text{L}^{2}$ Spaces, Berlin: Springer. 1978.

[18]

J. W. Hanneken, B. N. Narahari Achar and D. M. Vaught, An alpha–beta phase diagram representation of the zeros and properties of the Mittag–Leffler function, Adv. Math. Phys., (2013), Art. ID 421685, 13 pp. doi: 10.1155/2013/421685.

[19]

K. ItoB. Jin and T. Takeuchi, On the sectorial property of the Caputo derivative operator, Appl. Math. Lett., 47 (2015), 43-46.  doi: 10.1016/j.aml.2015.03.001.

[20]

B. JinR. LazarovJ. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.  doi: 10.1090/mcom/2960.

[21]

B. Jin and W. Rundell, An inverse Sturm-Liouville problem with a fractional derivative, J. Comput. Phys., 231 (2012), 4954-4966.  doi: 10.1016/j.jcp.2012.04.005.

[22]

A. P. Khromov, Finite–dimensional perturbations of Volterra operators, J. Math. Sci. (N.Y.), 138 (2006), 5893-6066.  doi: 10.1007/s10958-006-0346-9.

[23]

A. M. Nakhushev, The Sturm-Liouville problem for an ordinary second–order differential equation with fractional derivatives in the lowest terms, Soviet Mathematics, Doklady, 18 (1977), 666-670. 

[24]

T. Namba, P. Rybka and V. R. Voller, Some comments on using fractional derivative operators in modeling non-local diffusion processes, J. Comput. Appl. Math., 381 (2021), Paper No. 113040, 17 pp. doi: 10.1016/j.cam.2020.113040.

[25]

K. Oprzȩdkiewicz, Non integer order, state space model of heat transfer process using Caputo–Fabrizio operator, Bulletin of the Polish Academy of Sciences, Technical Sciences, 66 (2018), 249-255. 

[26]

K. Oprzȩdkiewicz and K. Dziedzic, New parameter identification method for the fractional order, state space model of heat transfer process, Automation, (2018), AISC 743,401–417.

[27]

K. Oprzȩdkiewicz and W. Mitkowski, A memory–efficient noninteger-order discrete time state-space model of a heat transfer process, Int. J. Appl. Math. Comput. Sci., 28 (2018), 49-659.  doi: 10.2478/amcs-2018-0050.

[28]

K. OprzȩdkiewiczW. MitkowskiK. Gawin and K. Dziedzic, The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process, Bulletin of the Polish Academy of Sciences, Technical Sciences, 66 (2018), 501-507. 

[29]

R. T. Parovik, An algorithm for computing the Mittag–Leffler function in the Maple symbolic mathematical package, International Journal of Soft Computing, 11 (2016), 487-491. 

[30]

M. Riesz, Intégrales de Riemann–Liouville et potentiels, Acta Universitatis Szegediensis, 9 (1938), 1–42,116–118.

[31]

K. Ryszewska, An analytic semigroup generated by a fractional differential operator, J. Math. Anal. Appl., 483 (2020), 123654, 17 pp. doi: 10.1016/j.jmaa.2019.123654.

[32]

K. Schmüdgen, Unbounded Self–adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, 265. Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.

[33]

S. TatarR. Tinaztepe and S. Ulusoy, Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal., 95 (2016), 1-23.  doi: 10.1080/00036811.2014.984291.

[34]

J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, 68. Springer-Verlag, New York-Berlin, 1980.

[35]

Q. Yang, Novel analytic and numerical methods for solving fractional dynamical systems, Ph.D. Thesis, School of Mathematical Sciences, Queensland University of Technology, (2010), 1–201, https://eprints.qut.edu.au/35750/1/Qianqian_Yang_Thesis.pdf.

Figure 3.1.  Plot of $ E_{\frac{3}{2}}(-z) $ over $ (-1, 2) $ – no real zeros
Figure 3.2.  Plot of $ E_{\frac{3}{2}}(-z) $ over $ (2, 60) $$ 3 $ real zeros detected
Figure 3.3.  Sector $ |\arg z|\leq\frac{3\pi }{4} $ and asymptotic zeros of $ E_{\frac{3}{2}} $
Figure 3.4.  Real zeros and the first 7 pairs of conjugate zeros of $ E_{\frac{3}{2}} $
Figure 3.5.  Image of the circle $ |z|\leq 36 $ under $ E_{\frac{3}{2}} $; $ \mathbf{3} $ encirclements of $ 0 $
Figure 3.6.  Image of the circle $ |z|\leq 135 $ under $ E_{\frac{3}{2}} $; $ \mathbf{5} $ encirclements of $ 0 $
Figure 4.1.  The heating of a rod system
Figure 4.2.  Plots of $ d $ (upper red curve) and $ d' $ (lower green curve) for $ \mu = 29.54150417 $ and $ \alpha = \frac{3}{2} $
Table 1.1.  Known and new results in $ \text{L}^{2}(0, 1) $
ItemOperator Its closure Generation of a.s. Generation of EXS a.s.
1a) $ J^{2-\alpha }\mathcal{A}_{1} $b) $ -RJ^{2-\alpha }L $c) Yes [31]d) Proved in Section 4.1e)
2f) $ J^{2-\alpha }\mathcal{A}_{DD} $g) $ _{\text{0}}^{C}{D}_{\theta }^{\alpha } $h) Yes [19]i) Proved in Section 4.2j)
3k) $ J^{2-\alpha }\mathcal{A}_{\mathfrak{RC}} $l) $ -RJ^{2-\alpha }LP $m) Proved in Section 4.3n)
4o) (4.19) (4.19) Proved in Section 4.4p)
5q) $ J^{n-\alpha }\mathcal{A}_{BC} $ $ \overline{J^{n-\alpha }\mathcal{A}_{BC}} $ Proposition 2.1r)
a) Neumann–Dirichlet boundary conditions.
b) $ \mathcal{A}_{1}=-LR $ – see (1.3).
c) Here with $ \gamma =\alpha -1 $ we have
ItemOperator Its closure Generation of a.s. Generation of EXS a.s.
1a) $ J^{2-\alpha }\mathcal{A}_{1} $b) $ -RJ^{2-\alpha }L $c) Yes [31]d) Proved in Section 4.1e)
2f) $ J^{2-\alpha }\mathcal{A}_{DD} $g) $ _{\text{0}}^{C}{D}_{\theta }^{\alpha } $h) Yes [19]i) Proved in Section 4.2j)
3k) $ J^{2-\alpha }\mathcal{A}_{\mathfrak{RC}} $l) $ -RJ^{2-\alpha }LP $m) Proved in Section 4.3n)
4o) (4.19) (4.19) Proved in Section 4.4p)
5q) $ J^{n-\alpha }\mathcal{A}_{BC} $ $ \overline{J^{n-\alpha }\mathcal{A}_{BC}} $ Proposition 2.1r)
a) Neumann–Dirichlet boundary conditions.
b) $ \mathcal{A}_{1}=-LR $ – see (1.3).
c) Here with $ \gamma =\alpha -1 $ we have
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