Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 61.

Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018066
Classification : 93E20, 60H30, 76A10, 49J20
Mots-clés : Relaxed controls, Young measure, hereditary evolution equations, martingale solution, Oldroyd fluid, Jeffreys fluid
Manna, Utpal 1 ; Mukherjee, Debopriya 1

1 
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Manna, Utpal; Mukherjee, Debopriya. Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 61. doi : 10.1051/cocv/2018066. http://www.numdam.org/articles/10.1051/cocv/2018066/

[1] N. Agram and B. Oksendal, Malliavin calculus and optimal control of stochastic Volterra equations. J. Optim. Theory Appl. 167 (2015) 1070–1094. | DOI | MR | Zbl

[2] S. Aizicovici and V. Barbu, Existence and asymptotic results for a system of integro-partial differential equations. NoDEA Nonlin. Differ. Equ. Appl. 3 (1996) 1–18. | DOI | MR | Zbl

[3] S. Aizicovici, M. Grasselli and M. Mckibben, A hyperbolic integrodifferential system related to phase-field models. Adv. Math. Sci. Appl. 10 (2000) 601–625. | MR | Zbl

[4] S. Albeverio, Z. Brzeźniak and J.L. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. 371 (2010) 309–322. | DOI | MR | Zbl

[5] D. Aldous, Stopping times and tightness. Ann. Probab. 6 (1978) 335–340. | DOI | MR | Zbl

[6] E.J. Balder, Lectures on Young measure theory and its applications in economics. Rend. Inst. Mat. Univ. Trieste 31 (2000) 1–69. | MR | Zbl

[7] V. Barbu, S. Bonaccorsi and L. Tubaro, Existence and asymptotic behavior for hereditary stochastic evolution equations. Appl. Math. Optim. 69 (2014) 273–314. | DOI | MR | Zbl

[8] V. Barbu, Z. Brzeźniak, E. Hausenblas, L. Tubaro, Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise. Stoch. Process. Appl. 123 (2013) 934–951. | DOI | MR | Zbl

[9] P. Billingsley, Convergence of Probability Measures. Wiley, New York (1999). | DOI | MR | Zbl

[10] S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control for stochastic Volterra equations with completely monotone kernels. SIAM J. Control Optim. 50 (2012) 748–789. | DOI | MR | Zbl

[11] S. Bonaccorsi and W. Desch, Volterra equations perturbed by noise. NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 557–594. | MR | Zbl

[12] H. Brezis. Analyse Fonctionnelle. Masson, Paris (1983). | MR | Zbl

[13] Z. Brzeźniak, E. Hausenblas and P.A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes. Potential Anal. 49 (2018) 131–201. | DOI | MR | Zbl

[14] Z. Brzeźniak, E. Hausenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise. Nonlinear Anal. 79 (2013) 122–139. | DOI | MR | Zbl

[15] Z. Brzeźniak and E. Motyl, Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D domains. J. Differ. Equ. 254 (2013) 1627–1685. | DOI | MR | Zbl

[16] Z. Brzeźniak, E. Motyl and M. Ondrejàt, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains. Ann. Probab. 45 (2017) 3145–3201. | DOI | MR | Zbl

[17] Z. Brzeźniak and M. Ondreját, Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces. Ann. Probab. 41 (2013) 1938–1977. | DOI | MR | Zbl

[18] Z. Brzeźniak and R. Serrano, Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces. SIAM J. Control Optim. 51 (2013) 2664–2703. | DOI | MR | Zbl

[19] C. Castaing, P. Raynaud De Fitte and M. Valadier, Young Measures on Topological Spaces: With Applications in Control Theory and Probability Theory. Vol. 571 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands (2004). | MR | Zbl

[20] M.-H. Chang, Stochastic control of hereditary systems and applications. In Vol. 59 of Stochastic Modelling and Applied Probability. Springer, New York (2008). | DOI | MR | Zbl

[21] Ph. Clément and G. Da Prato, White noise perturbation of the heat equation in materials with memory. Dyn. Syst. Appl. 6, (1997) 441–460. | MR | Zbl

[22] F. Confortola and E. Mastrogiacomo, Optimal control for stochastic heat equation with memory. Evol. Equ. Control Theory 3 (2014) 35–58. | DOI | MR | Zbl

[23] H. Crauel, Random Probability Measures on Polish Spaces. Vol. 11 of Stochastics Monographs. Taylor & Francis, London (2002). | MR | Zbl

[24] S.C. Crow, Viscoelastic properties of fine-grained incompressible turbulence. J. Fluid Mech. 33 (1968) 1–20. | DOI | Zbl

[25] N.J. Cutland and K. Grzesiak, Optimal control for 3D stochastic Navier-Stokes equations. Stochastics 77 (2005) 437–454. | DOI | MR | Zbl

[26] N.J. Cutland and K. Grzesiak, Optimal control for two-dimensional stochastic Navier-Stokes equations. Appl. Math. Optim. 55 (2007) 61–91. | DOI | MR | Zbl

[27] A. Doubova and E. Fernández-Cara, On the control of viscoelastic Jeffreys fluids. Syst. Control Lett. 61 (2012) 573–579. | DOI | MR | Zbl

[28] N.H. El Karoui, D. Nguyen and M. Jeanblanc-Picqué, Compactification methods in the control of degenerate diffusions: existence of an optimal control. Stochastics 20 (1987) 169–219. | DOI | MR | Zbl

[29] W.H. Fleming, Measure-valued processes in the control of partially-observable stochastic systems. Appl. Math. Optim. 6 (1980) 271–285. | DOI | MR | Zbl

[30] W.H. Fleming and M. Nisio, On stochastic relaxed control for partially observed diffusions. Nagoya Math. J. 93 (1984) 71–108. | DOI | MR | Zbl

[31] J. Fort and V. Méndez, Wavefront in time-delayed reaction-diffusion systems. Theory and comparison to experiments. Rep. Prog. Phys. 65 (2002) 895–954. | DOI

[32] D. Gatarek and J. Sobczyk, On the existence of optimal controls of Hilbert space-valued diffusions. SIAM J. Control Optim. 32 (1994) 170–175. | DOI | MR | Zbl

[33] C.H. Gibson, Turbulence in the ocean, atmosphere, galaxy, and universe. Appl. Mech. Rev. 49 (1996) 299–315. | DOI

[34] C. Giorgi and V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory. NoDEA Nonlinear Differ. Equ. Appl. 8 (2001) 157–171. | DOI | MR | Zbl

[35] M. Grasselli and V. Pata, A reaction-diffusion equation with memory. Discrete Continuous Dyn. Syst. 15 (2006) 1079–1088. | DOI | MR | Zbl

[36] M. Grasselli and V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory. Indiana Univ. Math. J. 50 (2001) 1281–1308. | DOI | MR | Zbl

[37] I. Gyöngy, On stochastic equations with respect to semimartingales III. Stochastics 7 (1982) 231–254. | DOI | MR | Zbl

[38] U.G. Haussmann and J.-P. Lepeltier, On the existence of optimal controls. SIAM J. Control Optim. 28 (1990) 851–902. | DOI | MR | Zbl

[39] J. Jacod and J. Memin, Sur un type de convergence intermediaire entre la convergence en loi et la convergence en probabilite. Seminaire de Probabilites XV, edited by J. Azema and M. Yor. Vol. 850 of Lecture Notes in Mathematics. Springer-Verlag, New York (1979/1980) 529–546. | DOI | Numdam | MR | Zbl

[40] A. Jakubowski, The almost sure Skorohod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen. 42 (1997) 209–216. [Translation in Theory Probab. Appl. 42 (1998) 167–174]. | DOI | MR | Zbl

[41] A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18 (1986) 20–65. | DOI | MR | Zbl

[42] D.D. Joseph, Fluid dynamics of Viscoelastic liquids. Springer Verlag, New York (1990). | DOI | MR | Zbl

[43] I. Klapper, C.J. Rupp, R. Cargo, B. Purvedorj and P. Stoodley, Viscoelastic fluid description of bacterial biofilm material properties. Biotechnol. Bioeng. 80 (2002) 289–296. | DOI

[44] C. Kuratowski, Topologie, Vol. I, 3rd edn, Monografie Matematyczne XX. Polskie Towarzystwo Matematyczne, Warsawa (1952). | MR | Zbl

[45] J.L. Lions, Optimal control of systems governed by partial differential equations. Translated from the French by S. K. Mitter. In Vol. 170 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York, Berlin (1971). | MR | Zbl

[46] A. Lunardi, On the linear heat equation with fading memory. SIAM J. Math. Anal. 21 (1990) 1213–1224. | DOI | MR | Zbl

[47] U. Manna, M.T. Mohan and S.S. Sritharan, Stochastic non-resistive magnetohydrodynamic system with Lévy noise. Random Oper. Stoch. Equ. 25 (2017) 155–193. | DOI | MR | Zbl

[48] C. Marinelli and M. Röckner, On the Maximal inequalities of Burkholder, Davis and Gundy. Expo. Math. 34 (2016) 1–26. | DOI | MR | Zbl

[49] M. Métivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces. Quaderni, Scuola Normale Superiore, Pisa (1988). | MR | Zbl

[50] R. Mikulevicius and B.L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations. Ann. Probab. 33 (2005) 137–176. | DOI | MR | Zbl

[51] R.K. Miller, Linear Volterra integrodifferntial equations as semigroups. Funkcial. Ekvac. 17 (1974) 39–55. | MR | Zbl

[52] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1. The MIT Press, MA (1971). | MR | Zbl

[53] E. Motyl, Martingale Solutions to the 2D and 3D Stochastic Navier-Stokes Equations Driven by the Compensated Poisson Random Measure, Preprint 13. Department of Mathematics and Computer Sciences, Lodz University (2011).

[54] E. Motyl, Stochastic Navier–Stokes equations driven by Lévy noise in unbounded 3D domains. Potential Anal. 38 (2013) 863–912. | MR | Zbl

[55] E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains-Abstract framework and applications. Stoch. Process. Appl. 124 (2014) 2052–2097. | DOI | MR | Zbl

[56] N. Nagase and M. Nisio, Optimal controls for stochastic partial differential equations. SIAM J. Control Optim. 28 (1990) 186–213. | DOI | MR | Zbl

[57] J.G. Oldroyd, Non-Newtonian flow of liquids and solids. In Vol. I of Rheology: Theory and applications, edited by F.R. Eirich. Academic Press Inc., New York, (1956), 653–682. | MR

[58] M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces. Diss. Math. 426 (2004) 1–63. | MR | Zbl

[59] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 (1979) 127–167. | DOI | MR | Zbl

[60] E. Pardoux, Équations aux dérivées partielles stochastiques de type monotone. Séminaire Jean Leray 3 (1975) 1–10. | Numdam | MR | Zbl

[61] K.R. Parthasarathy, Probability Measures on Metric Spaces. Academic Press, New York (1967). | MR | Zbl

[62] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. Vol. 13 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2007). | MR | Zbl

[63] J. Prüss, On linear volterra equations of parabolic type in Banach spaces. Trans. Am. Math. Soc. 301 (1987) 691–721. | DOI | MR | Zbl

[64] J. Prüss, Evolutionary Integral Equations and Applications. Vol. 87 of Monographs in Mathematics. Birkhaüser Verlag, Basel (1993). | DOI | MR | Zbl

[65] M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity. Vol. 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific and Technical, Harlow; John Wiley and Sons, Inc., New York (1987). | MR | Zbl

[66] R.S. Rivlin, The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids. Quart. Appl. Math. 15 (1957) 212–215. | DOI | MR | Zbl

[67] K. Sakthivel and S.S. Sritharan, Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evol. Equ. Control Theory 1 (2012) 355–392. | DOI | MR | Zbl

[68] S.E. Spagnolie, Complex Fluids in Biological Systems: Experiment, Theory, and Computation. Biological and Medical Physics Biomedical Engineering. Springer, New York (2015). | DOI | MR | Zbl

[69] S.S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyperviscosities. Appl Math. Optim. 41 (2000) 255–308. | DOI | MR | Zbl

[70] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Publishing, North-Holland, Amsterdam (1984). | MR | Zbl

[71] B.W. Towler, C.J. Rupp, A.B. Cunningham and P. Stoodley, Viscoelastic properties of a mixed culture biofilm from rheometer creep analysis. Biofouling 19 (2003) 279–285. | DOI

[72] F. Treves, Topological Vector Spaces, Distributions and Kernels. Academic Press, Indiana (1967). | MR | Zbl

[73] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). | DOI | MR | Zbl

[74] C.A. Truesdell and K.R. Rajagopal, An Introduction to the Mechanics of Fluids. Birkhauser, Boston (1999). | MR | Zbl

[75] E.O. Tuck, On positivity of Fourier transforms. Bull. Aust. Math. Soc. 74 (2006) 133–138. | DOI | MR | Zbl

[76] M. Valadier, Young measures, in Methods of Nonconvex Analysis (Varenna, 1989). Vol. 1446 of Lecture Notes in Mathematics. Springer, Berlin (1990) 152–188. | MR | Zbl

[77] J. Warga, Relaxed variational problems. J. Math. Anal. Appl. 4 (1962) 111–128. | DOI | MR | Zbl

[78] J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | MR | Zbl

[79] L.C. Young, Generalized surfaces in the calculus of variations. Ann. Math. 43 (1942) 84–103. | DOI | JFM | MR | Zbl

[80] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders, Philadelphia (1969). | MR | Zbl

[81] X.Y. Zhou, On the existence of optimal relaxed controls of stochastic partial differential equations. SIAM J. Control Optim. 30 (1992) 247–261. | DOI | MR | Zbl

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