doi: 10.3934/dcdss.2022079
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Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

1. 

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia, Spain

2. 

Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Mexico, Av. Universidad 940, C.P. 20131, Aguascalientes, Mexico

* Corresponding author: jccortes@imm.upv.es

Received  November 2021 Revised  February 2022 Early access March 2022

We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.

Citation: Juan C. Cortés, Sandra E. Delgadillo-Alemán, Roberto A. Kú-Carrillo, Rafael J. Villanueva. Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022079
References:
[1]

E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, Dordrecht, 2007.

[2]

H. T. Banks and S. Hu, Nonlinear stochastic Markov processes and modeling uncertainty in populations, Math. Biosci. Eng., 9 (2012), 1-25.  doi: 10.3934/mbe.2012.9.1.

[3]

C. Braumann, Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance, Wiley, 2019. doi: 10.1002/9781119166092.

[4]

S. Bunimovich-MendrazitskyH. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076.  doi: 10.1007/s11538-008-9344-z.

[5]

C. Burgos, J.-C. Cortés, L. Villafuerte and R.-J. Villanueva, Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density, J. Comput. Appl. Math., 378 (2020), 112925, 14 pp. doi: 10.1016/j.cam.2020.112925.

[6]

T. CaraballoJ.-C. Cortés and A. Navarro-Quiles, Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay, Appl. Math. Comput., 356 (2019), 198-218.  doi: 10.1016/j.amc.2019.03.048.

[7]

G. Casella and R. L. Berger, Statistical Inference, Cengage Learning, 2007.

[8]

M. S. CecconelloF. A. Dorini and G. Haeser, On fuzzy uncertainties on the logistic equation, Fuzzy Sets and Systems, 328 (2017), 107-121.  doi: 10.1016/j.fss.2017.07.011.

[9]

G. Chowell and H. Nishiura, Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Medicine, 12 (2014), Article number: 196, 17 pp. doi: 10.1186/s12916-014-0196-0.

[10]

J.-C. Cortés, S. Delgadillo-Alemán, R. Kú-Carrillo and R.-J. Villanueva, Full probabilistic analysis of random first-order linear differential equations with Dirac delta impulses appearing in control, Mathematical Methods in the Applied Sciences.

[11]

J.-C. Cortés, S. Delgadillo-Alemán, R. A. Kú-Carrillo and R.-J. Villanueva, Probabilistic analysis of a class of impulsive linear random differential equations via density functions, Appl. Math. Lett., 121 (2021), 107519, 9 pp. doi: 10.1016/j.aml.2021.107519.

[12]

J. CortésL. Jódar and L. Villafuerte, Mean square numerical solution of random differential equations: Facts and possibilities, Comput. Math. Appl., 53 (2007), 1098-1106.  doi: 10.1016/j.camwa.2006.05.030.

[13]

J.-C. CortésA. Navarro-QuilesJ.-V. Romero and M.-D. Roselló, Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty, J. Comput. Appl. Math., 337 (2018), 190-208.  doi: 10.1016/j.cam.2018.01.015.

[14]

F. A. DoriniN. Bobko and L. B. Dorini, A note on the logistic equation subject to uncertainties in parameters, Comput. Appl. Math., 37 (2018), 1496-1506.  doi: 10.1007/s40314-016-0409-6.

[15]

F. A. DoriniM. S. Cecconello and L. B. Dorini, On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density, Commun. Nonlinear Sci. Numer. Simul., 33 (2016), 160-173.  doi: 10.1016/j.cnsns.2015.09.009.

[16]

A. El FathiM. R. SmaouiV. GingrasB. Boulet and A. Haidar, The artificial pancreas and meal control: An overview of postprandial glucose regulation in type 1 diabetes, IEEE Control Syst., 38 (2018), 67-85.  doi: 10.1109/MCS.2017.2766323.

[17]

L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, New York, 2013. doi: 10.1090/mbk/082.

[18]

A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Mathematics and Its Applications, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9.

[19]

P. Georgescu and G. Moroșanu, Impulsive perturbations of a three-trophic prey-dependent food chain system, Math. Comput. Modelling, 48 (2008), 975-997.  doi: 10.1016/j.mcm.2007.12.006.

[20]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer Nature, 2017. doi: 10.1007/978-981-10-6265-0.

[21]

A. Hussein and M. M. Selim, Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique, Appl. Math. Comput., 218 (2012), 7193-7203.  doi: 10.1016/j.amc.2011.12.088.

[22]

P. E. Kloeden and E. Platen, Numerical Solution of Sstochastic Differential Equations, vol. 23, 3rd edition, Applications of Mathematics: Stochastic Modelling and Applied Probability, Springer, New York, 1999.

[23]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. doi: 10.1142/0906.

[24]

X. Li and P. Li, Input-to-state stability of nonlinear systems: Event-triggered impulsive control, IEEE Transactions on Automatic Control, 67 (2022), 1460-1465.  doi: 10.1109/TAC.2021.3063227.

[25]

X. Li and S. Song, Impulsive Systems with Delay. Stability and Control, Springer, Singapore, 2022. doi: 10.1007/978-981-16-4687-4.

[26]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[27]

X. LiangY. PeiM. Zhu and Y. Lv, Multiple kinds of optimal impulse control strategies on plant–pest–predator model with eco-epidemiology, Appl. Math. Comput., 287/288 (2016), 1-11.  doi: 10.1016/j.amc.2016.04.034.

[28]

M. Loève, Probability Theory I, Springer, New York, 1977.

[29]

G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, vol. 50, Cambridge University Press, 2014. doi: 10.1017/CBO9781139017329.

[30]

X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester,, 2008. doi: 10.1533/9780857099402.

[31]

T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Versita, London, 2013. doi: 10.2478/9788376560267.

[32]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, New York, 2010.

[33]

L. Shaikhet, Lyapunov Functional and Stability of Stochastic Differential Equations, Springer, 2013. doi: 10.1007/978-3-319-00101-2.

[34]

T. T. Soong, Random Differential Equations in Science and Engineering, Mathematics in Science and Engineering, Academic Press, Inc., New York, 1973.

[35]

A. VinodkumarM. Gowrisankar and P. Mohankumar, Existence, uniqueness and stability of random impulsive neutral partial differential equations, J. Egyptian Math. Soc., 23 (2015), 31-36.  doi: 10.1016/j.joems.2014.01.005.

[36]

S. Wu and Y. Duan, Oscillation stability and boundedness of second-order differential systems with random impulses, Comput. Math. Appl., 49 (2005), 1375-1386.  doi: 10.1016/j.camwa.2004.12.009.

[37]

S. Wu and X. Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 147-154.  doi: 10.1007/s10255-004-0157-z.

[38]

S. Zhang and J. Sun, Stability analysis of second-order differential systems with Erlang distribution random impulses, Adv. Difference Equ., 2013 (2013), 4, 10 pp. doi: 10.1186/1687-1847-2013-4.

show all references

References:
[1]

E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, Dordrecht, 2007.

[2]

H. T. Banks and S. Hu, Nonlinear stochastic Markov processes and modeling uncertainty in populations, Math. Biosci. Eng., 9 (2012), 1-25.  doi: 10.3934/mbe.2012.9.1.

[3]

C. Braumann, Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance, Wiley, 2019. doi: 10.1002/9781119166092.

[4]

S. Bunimovich-MendrazitskyH. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076.  doi: 10.1007/s11538-008-9344-z.

[5]

C. Burgos, J.-C. Cortés, L. Villafuerte and R.-J. Villanueva, Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density, J. Comput. Appl. Math., 378 (2020), 112925, 14 pp. doi: 10.1016/j.cam.2020.112925.

[6]

T. CaraballoJ.-C. Cortés and A. Navarro-Quiles, Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay, Appl. Math. Comput., 356 (2019), 198-218.  doi: 10.1016/j.amc.2019.03.048.

[7]

G. Casella and R. L. Berger, Statistical Inference, Cengage Learning, 2007.

[8]

M. S. CecconelloF. A. Dorini and G. Haeser, On fuzzy uncertainties on the logistic equation, Fuzzy Sets and Systems, 328 (2017), 107-121.  doi: 10.1016/j.fss.2017.07.011.

[9]

G. Chowell and H. Nishiura, Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Medicine, 12 (2014), Article number: 196, 17 pp. doi: 10.1186/s12916-014-0196-0.

[10]

J.-C. Cortés, S. Delgadillo-Alemán, R. Kú-Carrillo and R.-J. Villanueva, Full probabilistic analysis of random first-order linear differential equations with Dirac delta impulses appearing in control, Mathematical Methods in the Applied Sciences.

[11]

J.-C. Cortés, S. Delgadillo-Alemán, R. A. Kú-Carrillo and R.-J. Villanueva, Probabilistic analysis of a class of impulsive linear random differential equations via density functions, Appl. Math. Lett., 121 (2021), 107519, 9 pp. doi: 10.1016/j.aml.2021.107519.

[12]

J. CortésL. Jódar and L. Villafuerte, Mean square numerical solution of random differential equations: Facts and possibilities, Comput. Math. Appl., 53 (2007), 1098-1106.  doi: 10.1016/j.camwa.2006.05.030.

[13]

J.-C. CortésA. Navarro-QuilesJ.-V. Romero and M.-D. Roselló, Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty, J. Comput. Appl. Math., 337 (2018), 190-208.  doi: 10.1016/j.cam.2018.01.015.

[14]

F. A. DoriniN. Bobko and L. B. Dorini, A note on the logistic equation subject to uncertainties in parameters, Comput. Appl. Math., 37 (2018), 1496-1506.  doi: 10.1007/s40314-016-0409-6.

[15]

F. A. DoriniM. S. Cecconello and L. B. Dorini, On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density, Commun. Nonlinear Sci. Numer. Simul., 33 (2016), 160-173.  doi: 10.1016/j.cnsns.2015.09.009.

[16]

A. El FathiM. R. SmaouiV. GingrasB. Boulet and A. Haidar, The artificial pancreas and meal control: An overview of postprandial glucose regulation in type 1 diabetes, IEEE Control Syst., 38 (2018), 67-85.  doi: 10.1109/MCS.2017.2766323.

[17]

L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, New York, 2013. doi: 10.1090/mbk/082.

[18]

A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Mathematics and Its Applications, Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9.

[19]

P. Georgescu and G. Moroșanu, Impulsive perturbations of a three-trophic prey-dependent food chain system, Math. Comput. Modelling, 48 (2008), 975-997.  doi: 10.1016/j.mcm.2007.12.006.

[20]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer Nature, 2017. doi: 10.1007/978-981-10-6265-0.

[21]

A. Hussein and M. M. Selim, Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique, Appl. Math. Comput., 218 (2012), 7193-7203.  doi: 10.1016/j.amc.2011.12.088.

[22]

P. E. Kloeden and E. Platen, Numerical Solution of Sstochastic Differential Equations, vol. 23, 3rd edition, Applications of Mathematics: Stochastic Modelling and Applied Probability, Springer, New York, 1999.

[23]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. doi: 10.1142/0906.

[24]

X. Li and P. Li, Input-to-state stability of nonlinear systems: Event-triggered impulsive control, IEEE Transactions on Automatic Control, 67 (2022), 1460-1465.  doi: 10.1109/TAC.2021.3063227.

[25]

X. Li and S. Song, Impulsive Systems with Delay. Stability and Control, Springer, Singapore, 2022. doi: 10.1007/978-981-16-4687-4.

[26]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[27]

X. LiangY. PeiM. Zhu and Y. Lv, Multiple kinds of optimal impulse control strategies on plant–pest–predator model with eco-epidemiology, Appl. Math. Comput., 287/288 (2016), 1-11.  doi: 10.1016/j.amc.2016.04.034.

[28]

M. Loève, Probability Theory I, Springer, New York, 1977.

[29]

G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, vol. 50, Cambridge University Press, 2014. doi: 10.1017/CBO9781139017329.

[30]

X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester,, 2008. doi: 10.1533/9780857099402.

[31]

T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Versita, London, 2013. doi: 10.2478/9788376560267.

[32]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, New York, 2010.

[33]

L. Shaikhet, Lyapunov Functional and Stability of Stochastic Differential Equations, Springer, 2013. doi: 10.1007/978-3-319-00101-2.

[34]

T. T. Soong, Random Differential Equations in Science and Engineering, Mathematics in Science and Engineering, Academic Press, Inc., New York, 1973.

[35]

A. VinodkumarM. Gowrisankar and P. Mohankumar, Existence, uniqueness and stability of random impulsive neutral partial differential equations, J. Egyptian Math. Soc., 23 (2015), 31-36.  doi: 10.1016/j.joems.2014.01.005.

[36]

S. Wu and Y. Duan, Oscillation stability and boundedness of second-order differential systems with random impulses, Comput. Math. Appl., 49 (2005), 1375-1386.  doi: 10.1016/j.camwa.2004.12.009.

[37]

S. Wu and X. Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 147-154.  doi: 10.1007/s10255-004-0157-z.

[38]

S. Zhang and J. Sun, Stability analysis of second-order differential systems with Erlang distribution random impulses, Adv. Difference Equ., 2013 (2013), 4, 10 pp. doi: 10.1186/1687-1847-2013-4.

Figure 1.  3D-graphical representation of the approximate 1-p.d.f, $ f^N_1(x, t) $, of the solution stochastic process of the random IVP (1) for different orders of truncation $ N = 1, 2, 3 $, and impulse applications at $ T_{i}^{\delta} = i \Delta T $, $ i = 1, 2, 3, $ with $ \Delta T = 1/3 $. Example 1
Figure 2.  2D-graphical representation of the approximate 1-p.d.f., $ f^N_1(x, t) $, of the solution stochastic process of the random IVP (4) for different orders of truncation, $ N = 1, 2, 3, 5 $, and impulse applications at $ T_{i}^{\delta} = i \Delta T $, $ i = 1, 2, 3, $ with $ \Delta T = 1/3 $ at the times instants: $ t = 1/4, $ $ 1/2, $ $ 3/4. $ Example 1
Figure 3.  Comparison of the approximations of expectation, $ \mu_x^N $ (left panel) and standard deviation, $ \sigma_x^N $ (central panel), of the solution stochastic process using different orders of truncation, $ N = 1, 2, 3 $, for $ t \in [0, 1] $. Confidence intervals constructed according to the 2$ \sigma $-rule are shown taking $ N = 3 $, $ [\mu_x^3-2\sigma_x^3, \mu_x^3+2\sigma_x^3] $ (right panel). Example 1
Figure 4.  3D-graphical representation of the approximate 1-p.d.f, $ f^N_1(x, t) $, of the solution stochastic process of the random IVP (4) for different orders of truncation $ N = 1, 2, 3 $, and impulses at $ T_{i}^{\delta} = i \Delta T $, $ i = 1, 2, 3, $ with $ \Delta T = 1/3 $. Example 2
Figure 5.  2D-graphical representation of the approximate 1-p.d.f., $ f^N_1(x, t) $, of the solution stochastic process of the random IVP (4) for different orders of truncation, $ N = 1, 2, 3, 5 $, and impulses at $ T_{i}^{\delta} = i \Delta T $, $ i = 1, 2, 3, $ with $ \Delta T = 1/3 $ at the times instants $ t = 1/4, $ $ 1/2, $ $ 3/4. $ Example 2
Figure 6.  Comparison of the approximations of expectation, $ \mu_x^N $ (left panel) and standard deviation, $ \sigma_x^N $ (central panel), of the solution stochastic process using different orders of truncation $ N = 1, 2, 3 $, for $ t \in [0, 1] $. Confidence intervals, constructed according to the 2$ \sigma $-rule, are shown taking $ N = 3 $, $ [\mu_x^3-2\sigma_x^3, \mu_x^3+2\sigma_x^3] $ (right panel). Example 2
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