doi: 10.3934/eect.2021059
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Controlled singular evolution equations and Pontryagin type maximum principle with applications

1. 

Department of Mathematics, Xi'an Polytechnic University, Xi'an, Shaanxi 710048, China

2. 

CITMAga, Universidade de Vigo, Departamento de Matemática Aplicada II, E. E. Aeronáutica e do Espazo, Campus de Ourense, Universidade de Vigo, Ourense 32004, Spain

3. 

Departamento de Estatística, Análisis Matemático e Optimización, Instituto de Matemáticas, CITMAga, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

*Corresponding author: Xiao-Li Ding, dingding0605@126.com

Received  August 2020 Revised  August 2021 Early access December 2021

Fund Project: This work was supported by a key project of the International Science and Technology Cooperation Program of Shaanxi Research and Development Plan (2019KWZ-08). The work of J.J. Nieto has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain under Grant MTM2016-75140-P and co-financed by European Community fund FEDER and by Xunta de Galicia, grant ED431C 2019/02. This research was partially supported by the Portuguese Foundation for Science and Technology (FCT) within Project n. 147 Controlo Otimo e Modelacao Matematica da Pandemia COVID-19: contributos para uma estrategia sistemica de intervencao em saude na omunidade, in the scope of the RESEARCH 4 COVID-19 call financed by FCT; and by the Instituto de Salud Carlos III, within the Project COV20-00617 Prediccion dinamica de escenarios de afectacion por COVID-19 a corto y medio plazo (PREDICO), in the scope of the Fondo COVID financed by the Ministerio de Ciencia e Innovacion of Spain

Due to the propagation of new coronavirus (COVID-19) on the community, global researchers are concerned with how to minimize the impact of COVID-19 on the world. Mathematical models are effective tools that help to prevent and control this disease. This paper mainly focuses on the optimal control problems of an epidemic system governed by a class of singular evolution equations. The mild solutions of such equations of Riemann-Liouville or Caputo types are special cases of the proposed equations. We firstly discuss well-posedness in an appropriate functional space for such equations. In order to reduce the cost caused by control process and vaccines, and minimize the total number of susceptible people and infected people as much as possible, an optimal control problem of an epidemic system is presented. And then for associated control problem, we use a generalized Liapunov type theorem and the spike perturbation technique to obtain a Pontryagin type maximum principle for its optimal controls. In order to derive the maximum principle for an optimal control problems, some techniques from analytical semigroups are employed to overcome the difficulties. Finally, we discuss the potential applications.

Citation: Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, doi: 10.3934/eect.2021059
References:
[1]

M. S. AbdoK. ShahH. A. Wahash and S. K. Panchal, On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative, Chaos, Solitons Fractals, 135 (2020), 109867.  doi: 10.1016/j.chaos.2020.109867.

[2]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6.

[3]

O. P. AgrawalO. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables, J. Vib. Control, 16 (2010), 1967-1976.  doi: 10.1177/1077546309353361.

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A. AlsaediJ. J. Nieto and V. Venktesh, Fractional electrical circuits, Adv. in Mechanical Engineering, 7 (2015), 1-7. 

[5]

J. E. Anderson and H. F. Xu, Necessary and sufficient conditions for optimal offers in electricity markets, SIAM J. Control Optim., 41 (2002), 1212-1228.  doi: 10.1137/S0363012900367801.

[6]

L. M. Betz, Second-order sufficient optimality conditions for optimal control of nonsmooth, semilinear parabolic equations, SIAM J. Control Optim., 57 (2019), 4033-4062.  doi: 10.1137/19M1239106.

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V. I. Bogachev, Measure Theory, I, Springer, New York, 2007. doi: 10.1007/978-3-540-34514-5.

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L. Bourdin, A class of fractional optimal control problems and fractional Pontryagin's system. Existence of a fractional Noether's theorem, preprint, arXiv: 1203.1422v1, 2012.

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C. Burnap and M. A. Kazemi, Optimal control of a system governed by nonlinear Volterra integral equations with delay, IMA J. Math. Control Inform., 16 (1999), 73-89.  doi: 10.1093/imamci/16.1.73.

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D. A. Carlson, An elementay proof of the maximum principle for optimal control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 54 (1987), 43-61.  doi: 10.1007/BF00940404.

[11]

M. Dalir and N. Bigdeli, The design of a new hybrid controller for fractional-order uncertain chaotic systems with unknown time-varying delays, Applied Soft Computing, 87 (2020), 106000.  doi: 10.1016/j.asoc.2019.106000.

[12]

C. De La Vega, Necessary conditions for optimal terminal time control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 130 (2006), 79-93.  doi: 10.1007/s10957-006-9087-7.

[13]

X. L. Ding and Y. L. Jiang, Semilinear fractional differential equations based on a new integral operator approach, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 5143-5150.  doi: 10.1016/j.cnsns.2012.03.036.

[14]

K. Du and Q. X. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.  doi: 10.1137/120882433.

[15]

F. Dufour and B. Miller, Maximum principle for singular stochastic control problems, SIAM J. Control Optim., 45 (2006), 668-698.  doi: 10.1137/040612403.

[16]

A. FarhadiG. H. Erjaee and M. Salehi, Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Comput. Math. Appl., 73 (2017), 2066-2075.  doi: 10.1016/j.camwa.2017.02.031.

[17]

M. G. Hall and T. R. Barrick, From diffusion-weighted MRI to anomalous diffusion imaging, Magn. Reson. Med., 59 (2008), 447-455.  doi: 10.1002/mrm.21453.

[18]

S. HeS. Tang and L. Rong, A discrete stochastic model of the COVID-19 outbreak: Forecast and control, Math. Biosci. Eng., 17 (2020), 2792-2804.  doi: 10.3934/mbe.2020153.

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.

[20]

M. Higazy, Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic, Chaos, Solitons Fractals, 138 (2020), 110007.  doi: 10.1016/j.chaos.2020.110007.

[21]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[22]

R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.

[23]

R. Kamocki and M. Majewski, Fractional linear control systems with Caputo derivative and their optimization, Optim. Control Appl. Meth., 36 (2015), 953-967.  doi: 10.1002/oca.2150.

[24]

S. H. A. KhoshnawM. ShahzadM. Ali and F. Sultan, A quantitative and qualitative analysis of the COVID-19 pandemic model, Chaos Solitons Fractals, 138 (2020), 109932.  doi: 10.1016/j.chaos.2020.109932.

[25]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006.

[26]

G. S. Ladde and L. Wu, Development of nonlinear stochastic models by using stock price data and basic statistics, Neutral Parallel Sci. Comput., 18 (2010), 269-282. 

[27]

X. J. Li and J. M. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908.  doi: 10.1137/0329049.

[28]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.

[29]

P. Lin and J. M. Yong, Controlled singular Volterra integral equations and Pontryagin maximum principle, SIAM J. Control Optim., 58 (2020), 136-164.  doi: 10.1137/19M124602X.

[30]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.

[31] F. Mainardi, Fractional Calculus and Waves In Linear Viscoelasticity, London, Imperial College Press, 2010.  doi: 10.1142/9781848163300.
[32]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[33]

T. Y. MiyaokaS. Lenhart and J. F. C. A. Meyer, Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus, J. Math. Biol., 79 (2019), 1077-1104.  doi: 10.1007/s00285-019-01390-z.

[34]

F. NdaïrouI. AreaJ. J. Nieto and D. F. M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons Fractals, 135 (2020), 109846.  doi: 10.1016/j.chaos.2020.109846.

[35]

F. NdaïrouI. Area and J. J. Nieto, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos, Solitons Fractals, 144 (2021), 110652.  doi: 10.1016/j.chaos.2021.110652.

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon Publishers, 1993.

[38]

C. J. SilvaC. Cruz and D. F. M. Torres, Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal, Scientific Reports, 11 (2021), 3451.  doi: 10.1038/s41598-021-83075-6.

[39]

B. TangX. WangQ. LiN. L. BragazziS. TangY. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. 

[40]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[41]

K. Yosida, Functional Analysis, 6$^{th}$ edition, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.

[42]

H. S. Zhang and X. Zhang, Second-order necessary conditions for stochastic optimal control problems, SIAM Rev., 60 (2018), 139-178.  doi: 10.1137/17M1148773.

[43]

W. G. ZhouC. D. HuangM. Xiao and J. D. Cao, Hybrid tactics for bifurcation control in a fractional-order delayed predator-prey model, Phys. A, 515 (2019), 183-191.  doi: 10.1016/j.physa.2018.09.185.

[44]

M. ZhouH. L. Xiang and Z. X. Li, Optimal control strategies for a reaction-diffusion epidemic system, Nonlinear Anal. Real World Appl., 46 (2019), 446-464.  doi: 10.1016/j.nonrwa.2018.09.023.

show all references

References:
[1]

M. S. AbdoK. ShahH. A. Wahash and S. K. Panchal, On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative, Chaos, Solitons Fractals, 135 (2020), 109867.  doi: 10.1016/j.chaos.2020.109867.

[2]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6.

[3]

O. P. AgrawalO. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables, J. Vib. Control, 16 (2010), 1967-1976.  doi: 10.1177/1077546309353361.

[4]

A. AlsaediJ. J. Nieto and V. Venktesh, Fractional electrical circuits, Adv. in Mechanical Engineering, 7 (2015), 1-7. 

[5]

J. E. Anderson and H. F. Xu, Necessary and sufficient conditions for optimal offers in electricity markets, SIAM J. Control Optim., 41 (2002), 1212-1228.  doi: 10.1137/S0363012900367801.

[6]

L. M. Betz, Second-order sufficient optimality conditions for optimal control of nonsmooth, semilinear parabolic equations, SIAM J. Control Optim., 57 (2019), 4033-4062.  doi: 10.1137/19M1239106.

[7]

V. I. Bogachev, Measure Theory, I, Springer, New York, 2007. doi: 10.1007/978-3-540-34514-5.

[8]

L. Bourdin, A class of fractional optimal control problems and fractional Pontryagin's system. Existence of a fractional Noether's theorem, preprint, arXiv: 1203.1422v1, 2012.

[9]

C. Burnap and M. A. Kazemi, Optimal control of a system governed by nonlinear Volterra integral equations with delay, IMA J. Math. Control Inform., 16 (1999), 73-89.  doi: 10.1093/imamci/16.1.73.

[10]

D. A. Carlson, An elementay proof of the maximum principle for optimal control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 54 (1987), 43-61.  doi: 10.1007/BF00940404.

[11]

M. Dalir and N. Bigdeli, The design of a new hybrid controller for fractional-order uncertain chaotic systems with unknown time-varying delays, Applied Soft Computing, 87 (2020), 106000.  doi: 10.1016/j.asoc.2019.106000.

[12]

C. De La Vega, Necessary conditions for optimal terminal time control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 130 (2006), 79-93.  doi: 10.1007/s10957-006-9087-7.

[13]

X. L. Ding and Y. L. Jiang, Semilinear fractional differential equations based on a new integral operator approach, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 5143-5150.  doi: 10.1016/j.cnsns.2012.03.036.

[14]

K. Du and Q. X. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.  doi: 10.1137/120882433.

[15]

F. Dufour and B. Miller, Maximum principle for singular stochastic control problems, SIAM J. Control Optim., 45 (2006), 668-698.  doi: 10.1137/040612403.

[16]

A. FarhadiG. H. Erjaee and M. Salehi, Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Comput. Math. Appl., 73 (2017), 2066-2075.  doi: 10.1016/j.camwa.2017.02.031.

[17]

M. G. Hall and T. R. Barrick, From diffusion-weighted MRI to anomalous diffusion imaging, Magn. Reson. Med., 59 (2008), 447-455.  doi: 10.1002/mrm.21453.

[18]

S. HeS. Tang and L. Rong, A discrete stochastic model of the COVID-19 outbreak: Forecast and control, Math. Biosci. Eng., 17 (2020), 2792-2804.  doi: 10.3934/mbe.2020153.

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.

[20]

M. Higazy, Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic, Chaos, Solitons Fractals, 138 (2020), 110007.  doi: 10.1016/j.chaos.2020.110007.

[21]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[22]

R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.

[23]

R. Kamocki and M. Majewski, Fractional linear control systems with Caputo derivative and their optimization, Optim. Control Appl. Meth., 36 (2015), 953-967.  doi: 10.1002/oca.2150.

[24]

S. H. A. KhoshnawM. ShahzadM. Ali and F. Sultan, A quantitative and qualitative analysis of the COVID-19 pandemic model, Chaos Solitons Fractals, 138 (2020), 109932.  doi: 10.1016/j.chaos.2020.109932.

[25]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006.

[26]

G. S. Ladde and L. Wu, Development of nonlinear stochastic models by using stock price data and basic statistics, Neutral Parallel Sci. Comput., 18 (2010), 269-282. 

[27]

X. J. Li and J. M. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908.  doi: 10.1137/0329049.

[28]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.

[29]

P. Lin and J. M. Yong, Controlled singular Volterra integral equations and Pontryagin maximum principle, SIAM J. Control Optim., 58 (2020), 136-164.  doi: 10.1137/19M124602X.

[30]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.

[31] F. Mainardi, Fractional Calculus and Waves In Linear Viscoelasticity, London, Imperial College Press, 2010.  doi: 10.1142/9781848163300.
[32]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[33]

T. Y. MiyaokaS. Lenhart and J. F. C. A. Meyer, Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus, J. Math. Biol., 79 (2019), 1077-1104.  doi: 10.1007/s00285-019-01390-z.

[34]

F. NdaïrouI. AreaJ. J. Nieto and D. F. M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons Fractals, 135 (2020), 109846.  doi: 10.1016/j.chaos.2020.109846.

[35]

F. NdaïrouI. Area and J. J. Nieto, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos, Solitons Fractals, 144 (2021), 110652.  doi: 10.1016/j.chaos.2021.110652.

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon Publishers, 1993.

[38]

C. J. SilvaC. Cruz and D. F. M. Torres, Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal, Scientific Reports, 11 (2021), 3451.  doi: 10.1038/s41598-021-83075-6.

[39]

B. TangX. WangQ. LiN. L. BragazziS. TangY. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. 

[40]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[41]

K. Yosida, Functional Analysis, 6$^{th}$ edition, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.

[42]

H. S. Zhang and X. Zhang, Second-order necessary conditions for stochastic optimal control problems, SIAM Rev., 60 (2018), 139-178.  doi: 10.1137/17M1148773.

[43]

W. G. ZhouC. D. HuangM. Xiao and J. D. Cao, Hybrid tactics for bifurcation control in a fractional-order delayed predator-prey model, Phys. A, 515 (2019), 183-191.  doi: 10.1016/j.physa.2018.09.185.

[44]

M. ZhouH. L. Xiang and Z. X. Li, Optimal control strategies for a reaction-diffusion epidemic system, Nonlinear Anal. Real World Appl., 46 (2019), 446-464.  doi: 10.1016/j.nonrwa.2018.09.023.

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