• Previous Article
    Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity
  • EECT Home
  • This Issue
  • Next Article
    Impulsive hemivariational inequality for a class of history-dependent quasistatic frictional contact problems
doi: 10.3934/eect.2022015
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data

1. 

African Institute for Mathematical Sciences (AIMS), P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon, Institut für Analysis, Fakultät Mathematik, Technische Universität Dresden D-01062 Dresden, Germany

2. 

Laboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, (FWI), Guadeloupe, France

3. 

Department of Mathematical Sciences, Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, 4400 University Dr, Fairfax, VA 22030, USA

* Corresponding author: Mahamadi Warma

Received  March 2021 Revised  February 2022 Early access March 2022

Fund Project: The first author is supported by the Deutscher Akademischer Austausch Dienst/German Academic Exchange Service (DAAD). The third author is partially supported by the AFOSR under Award NO: FA9550-18-1-0242 and by the US Army Research Office (ARO) under Award NO: W911NF-20-1-0115

We consider parabolic equations on bounded smooth open sets $ {\Omega}\subset \mathbb{R}^N $ ($ N\ge 1 $) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $ \mathscr{L} : = - \Delta + (-\Delta)^{s} $ ($ 0<s<1 $). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been presented and studied in a unified fashion for mixed local-nonlocal PDEs with singular data.

Citation: Jean-Daniel Djida, Gisèle Mophou, Mahamadi Warma. Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data. Evolution Equations and Control Theory, doi: 10.3934/eect.2022015
References:
[1]

P. AcquistapaceF. Flandoli and B. Terreni, Initial-boundary value problems and optimal control for nonautonomous parabolic systems, SIAM J. Control Optim., 29 (1991), 89-118.  doi: 10.1137/0329005.

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. 
[3]

H. Antil and S. Bartels, Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.  doi: 10.1515/cmam-2017-0039.

[4]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 084003, 35 pp. doi: 10.1088/1361-6420/ab1299.

[5]

H. Antil, D. Verma and M. Warma, External optimal control of fractional parabolic PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), 33 pp. doi: 10.1051/cocv/2020005.

[6]

H. AntilD. Verma and M. Warma, Optimal control of fractional elliptic PDEs with state constraints and characterization of the dual of fractional-order Sobolev spaces, J. Optim. Theory Appl., 186 (2020), 1-23.  doi: 10.1007/s10957-020-01684-z.

[7]

H. Antil and M. Warma, Optimal control of fractional semilinear PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), 30 pp. doi: 10.1051/cocv/2019061.

[8]

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

[9]

W. ArendtA. F. M. ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differential Equations, 43 (2018), 1-24.  doi: 10.1080/03605302.2017.1363229.

[10]

S. BiagiS. DipierroE. Valdinoci and E. Vecchi, Mixed local and nonlocal elliptic operators: Regularity and maximum principle, Comm. Partial Differential Equations, 47 (2021), 585-629.  doi: 10.1080/03605302.2021.1998908.

[11]

S. BiagiE. VecchiS. Dipierro and E. Valdinoci, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), 1611-1641.  doi: 10.1017/prm.2020.75.

[12]

U. BiccariM. Warma and E. Zuazua, Local regularity for fractional heat equations, Recent Advances in PDEs: Analysis, Numerics and Control, SEMA SIMAI Springer Ser., Springer, Cham, 17 (2018), 233-249. 

[13]

K. BogdanK. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1.

[14]

J. P. BorthagarayD. Leykekhman and R. H. Nochetto, Local energy estimates for the fractional Laplacian, SIAM J. Numer. Anal., 59 (2021), 1918-1947.  doi: 10.1137/20M1335509.

[15]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[16]

A. Bueno-OrovioD. KayV. GrauB. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization, Journal of The Royal Society Interface, 11 (2014).  doi: 10.1098/rsif.2014.0352.

[17]

V. I. Burenkov, Sobolev Spaces on Domains, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 137. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4.

[18]

L. A. CaffarelliJ.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.

[19]

R. S. CantrellC. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.

[20]

W. Chen, A speculative study of $2/3$-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 023126.  doi: 10.1063/1.2208452.

[21]

B. Claus and M. Warma, Realization of the fractional Laplacian with nonlocal exterior conditions via forms method, J. Evol. Equ., 20 (2020), 1597-1631.  doi: 10.1007/s00028-020-00567-0.

[22] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1990. 
[23]

M. C. Delfour and M. Sorine, The linear-quadratic optimal control problem for parabolic systems with boundary control through a Dirichlet condition, Control of Distributed Parameter Systems, 1982 (Toulouse, 1982), IFAC, Luxenburg, (1983), 87-90. 

[24]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[25]

S. DipierroE. P. Lippi and E. Valdinoci, Linear theory for a mixed operator with Neumann conditions, Asymptotic Analysis Preprint, (2021), 1-24. 

[26]

S. DipierroX. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416.  doi: 10.4171/RMI/942.

[27]

R. DorvilleG. M. Mophou and V. Valmorin, Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1472-1481.  doi: 10.1016/j.camwa.2011.03.025.

[28]

A. FiscellaR. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.

[29]

C. G. Gal and M. Warma, Bounded solutions for nonlocal boundary value problems on Lipschitz manifolds with boundary, Adv. Nonlinear Stud., 16 (2016), 529-550.  doi: 10.1515/ans-2015-5033.

[30]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.

[31]

C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ., 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4.

[32]

F. Gesztesy and M. Mitrea, A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math., 113 (2011), 53-172.  doi: 10.1007/s11854-011-0002-2.

[33]

T. Ghosh, A. Rüland and M. Salo, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505, 42 pp. doi: 10.1016/j.jfa.2020.108505.

[34]

T. GhoshM. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.  doi: 10.2140/apde.2020.13.455.

[35]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[36]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, 69. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.

[37]

G. Grubb, Regularity in Lp Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), 2634-2660.  doi: 10.1016/j.jfa.2017.12.011.

[38]

I. Lasiecka, Boundary control of parabolic systems: Regularity of optimal solutions, Appl. Math. Optim., 4 (1977/78), 301-327.  doi: 10.1007/BF01442147.

[39]

I. Lasiecka and R. Triggiani, Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and Riccati's feedback synthesis, SIAM J. Control Optim., 21 (1983), 41-67.  doi: 10.1137/0321003.

[40] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. 
[41]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.

[42]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin 1971.

[43]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[44]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. Ⅱ, Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg, 1972.

[45]

V. G. Maz'ya and S. V. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997.

[46] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ, 2005. 
[47]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[48]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.

[49]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[50]

G. M. ViswanathanV. AfanasyevS. V. BuldyrevE. J. MurphyP. A. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415.  doi: 10.1038/381413a0.

[51]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4.

[52]

M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 1, 46 pp. doi: 10.1007/s00030-016-0354-5.

[53]

M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.  doi: 10.1137/18M117145X.

show all references

References:
[1]

P. AcquistapaceF. Flandoli and B. Terreni, Initial-boundary value problems and optimal control for nonautonomous parabolic systems, SIAM J. Control Optim., 29 (1991), 89-118.  doi: 10.1137/0329005.

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. 
[3]

H. Antil and S. Bartels, Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.  doi: 10.1515/cmam-2017-0039.

[4]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 084003, 35 pp. doi: 10.1088/1361-6420/ab1299.

[5]

H. Antil, D. Verma and M. Warma, External optimal control of fractional parabolic PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), 33 pp. doi: 10.1051/cocv/2020005.

[6]

H. AntilD. Verma and M. Warma, Optimal control of fractional elliptic PDEs with state constraints and characterization of the dual of fractional-order Sobolev spaces, J. Optim. Theory Appl., 186 (2020), 1-23.  doi: 10.1007/s10957-020-01684-z.

[7]

H. Antil and M. Warma, Optimal control of fractional semilinear PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), 30 pp. doi: 10.1051/cocv/2019061.

[8]

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

[9]

W. ArendtA. F. M. ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differential Equations, 43 (2018), 1-24.  doi: 10.1080/03605302.2017.1363229.

[10]

S. BiagiS. DipierroE. Valdinoci and E. Vecchi, Mixed local and nonlocal elliptic operators: Regularity and maximum principle, Comm. Partial Differential Equations, 47 (2021), 585-629.  doi: 10.1080/03605302.2021.1998908.

[11]

S. BiagiE. VecchiS. Dipierro and E. Valdinoci, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), 1611-1641.  doi: 10.1017/prm.2020.75.

[12]

U. BiccariM. Warma and E. Zuazua, Local regularity for fractional heat equations, Recent Advances in PDEs: Analysis, Numerics and Control, SEMA SIMAI Springer Ser., Springer, Cham, 17 (2018), 233-249. 

[13]

K. BogdanK. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1.

[14]

J. P. BorthagarayD. Leykekhman and R. H. Nochetto, Local energy estimates for the fractional Laplacian, SIAM J. Numer. Anal., 59 (2021), 1918-1947.  doi: 10.1137/20M1335509.

[15]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[16]

A. Bueno-OrovioD. KayV. GrauB. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization, Journal of The Royal Society Interface, 11 (2014).  doi: 10.1098/rsif.2014.0352.

[17]

V. I. Burenkov, Sobolev Spaces on Domains, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 137. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4.

[18]

L. A. CaffarelliJ.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.

[19]

R. S. CantrellC. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.

[20]

W. Chen, A speculative study of $2/3$-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 023126.  doi: 10.1063/1.2208452.

[21]

B. Claus and M. Warma, Realization of the fractional Laplacian with nonlocal exterior conditions via forms method, J. Evol. Equ., 20 (2020), 1597-1631.  doi: 10.1007/s00028-020-00567-0.

[22] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1990. 
[23]

M. C. Delfour and M. Sorine, The linear-quadratic optimal control problem for parabolic systems with boundary control through a Dirichlet condition, Control of Distributed Parameter Systems, 1982 (Toulouse, 1982), IFAC, Luxenburg, (1983), 87-90. 

[24]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[25]

S. DipierroE. P. Lippi and E. Valdinoci, Linear theory for a mixed operator with Neumann conditions, Asymptotic Analysis Preprint, (2021), 1-24. 

[26]

S. DipierroX. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416.  doi: 10.4171/RMI/942.

[27]

R. DorvilleG. M. Mophou and V. Valmorin, Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1472-1481.  doi: 10.1016/j.camwa.2011.03.025.

[28]

A. FiscellaR. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.

[29]

C. G. Gal and M. Warma, Bounded solutions for nonlocal boundary value problems on Lipschitz manifolds with boundary, Adv. Nonlinear Stud., 16 (2016), 529-550.  doi: 10.1515/ans-2015-5033.

[30]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.

[31]

C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ., 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4.

[32]

F. Gesztesy and M. Mitrea, A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math., 113 (2011), 53-172.  doi: 10.1007/s11854-011-0002-2.

[33]

T. Ghosh, A. Rüland and M. Salo, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505, 42 pp. doi: 10.1016/j.jfa.2020.108505.

[34]

T. GhoshM. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.  doi: 10.2140/apde.2020.13.455.

[35]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[36]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, 69. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.

[37]

G. Grubb, Regularity in Lp Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), 2634-2660.  doi: 10.1016/j.jfa.2017.12.011.

[38]

I. Lasiecka, Boundary control of parabolic systems: Regularity of optimal solutions, Appl. Math. Optim., 4 (1977/78), 301-327.  doi: 10.1007/BF01442147.

[39]

I. Lasiecka and R. Triggiani, Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and Riccati's feedback synthesis, SIAM J. Control Optim., 21 (1983), 41-67.  doi: 10.1137/0321003.

[40] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. 
[41]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.

[42]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin 1971.

[43]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[44]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. Ⅱ, Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg, 1972.

[45]

V. G. Maz'ya and S. V. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997.

[46] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ, 2005. 
[47]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[48]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.

[49]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[50]

G. M. ViswanathanV. AfanasyevS. V. BuldyrevE. J. MurphyP. A. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415.  doi: 10.1038/381413a0.

[51]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4.

[52]

M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 1, 46 pp. doi: 10.1007/s00030-016-0354-5.

[53]

M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.  doi: 10.1137/18M117145X.

[1]

Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657

[2]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control and Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022

[3]

J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431

[4]

Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control and Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007

[5]

Andrei V. Dmitruk, Alexander M. Kaganovich. Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 523-545. doi: 10.3934/dcds.2011.29.523

[6]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial and Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329

[7]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[8]

Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks and Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027

[9]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001

[10]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[11]

Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017

[12]

Jiongmin Yong. Optimality conditions for controls of semilinear evolution systems with mixed constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 371-388. doi: 10.3934/dcds.1995.1.371

[13]

William G. Litvinov. Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions. Journal of Industrial and Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291

[14]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101

[15]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

[16]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

[17]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070

[18]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505

[19]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174

[20]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control and Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (141)
  • HTML views (68)
  • Cited by (0)

[Back to Top]