doi: 10.3934/eect.2020110

Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints

1. 

Department of Mathematics, University of Maryland, College Park, MD 20740, USA

2. 

Department of Mathematics, Indian Institute of Technology Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

3. 

M. S. Ramaiah University of Applied Sciences, University House, New BEL Road, MSR Nagar, Bangalore-560 054a, India

*Corresponding author: doboss27@umd.edu

Received  September 2020 Revised  September 2020 Early access  December 2020

A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows subject to state constraints is established using the Ekeland variational principle. Since the system considered here is of coupled parabolic-hyperbolic type, the well developed control theory literature using abstract semigroup approach to linear and semilinear partial differential equations does not seem to contain problems of the type studied in this paper. The controls are distributed over a bounded domain, while the state variables are subject to a set of constraints and governed by the compressible Navier-Stokes equations linearized around a suitably regular base state. The maximum principle is of integral-type and obtained for minimizers of a tracking-type integral cost functional.

Citation: Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, doi: 10.3934/eect.2020110
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer-Verlag, 2006.  Google Scholar

[2]

E. V. Amosova, Optimal control of a viscous heat-conducting gas flow, Journal of Applied and Industrial Mathematics, 3 (2009), 5-20.  doi: 10.1134/S1990478909010025.  Google Scholar

[3]

V. Barbu, Optimal Control of Variational Inequalities, Pitman Advanced Publishing Program, 1984.  Google Scholar

[4]

V. Barbu, Mathematical Methods in Optimization in Differential Systems, Springer, 1994. doi: 10.1007/978-94-011-0760-0.  Google Scholar

[5]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar

[6]

P. BellaE. FeireislB. J. Jin and A. Novotný, Robustness of strong solutions to the compressible Navier–Stokes system, Mathematische Annalen, 362 (2015), 281-303.  doi: 10.1007/s00208-014-1119-2.  Google Scholar

[7]

J. Borggaard and J. Burns, A PDE sensitivity equation method for optimal aerodynamic design, Journal of Computational Physics, 136 (1997), 366-384.  doi: 10.1006/jcph.1997.5743.  Google Scholar

[8]

Y. Cho and H. Kim, On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities, Manuscripta Mathematica, 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[9]

S. Chowdhury and M. Ramaswamy, Optimal control of linearized compressible Navier–Stokes equations, ESAIM: Control, Optimization and Calculus of Variations, 19 (2013), 587-615.  doi: 10.1051/cocv/2012023.  Google Scholar

[10]

S. ChowdhuryM. Ramaswamy and J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier–Stokes system in one-dimension, SIAM Journal on Control and Optimization, 50 (2012), 2959-2987.  doi: 10.1137/110846683.  Google Scholar

[11]

S. S. Collis, K. Ghayour, M. Heinkenschloss, M. Ulbrich and S. Ulbrich, Towards adjoint-based methods for aeroacoustic control, in 39th Aerospace Science Meeting & Exhibit, Reno, NV, AIAA PAPER, (2001), 2001-821, 1–17. doi: 10.2514/6.2001-821.  Google Scholar

[12]

S. S. Collis, K. Ghayour, M. Heinkenschloss, M. Ulbrich and S. Ulbrich, Numerical solution of optimal control problems governed by the compressible Navier–Stokes equations, in Optimal Control of Complex Structures: International Conference in Oberwolfach, Birkhäuser Basel, Basel, (2002), 43–55.  Google Scholar

[13]

H. B. da Veiga, Diffusion on viscous fluids, Existence and asymptotic properties of solutions, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, 10 (1983), 341–355.  Google Scholar

[14]

S. DoboszczakM. T. Mohan and S. S. Sritharan, Existence of optimal controls for compressible viscous flow, Journal of Mathematical Fluid Mechanics, 20 (2018), 199-211.  doi: 10.1007/s00021-017-0318-5.  Google Scholar

[15]

I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications, 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

I. Ekeland, Nonconvex minimization problems, Bulletin of the American Mathematical Society, 1 (1979), 443-473.  doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar

[17]

S. ErvedozaO. GlassS. Guerrero and J.-P. Puel, Local exact controllability for the one–dimensional compressible Navier–Stokes equation, Arch. Rational Mech. Anal., 206 (2012), 189-238.  doi: 10.1007/s00205-012-0534-3.  Google Scholar

[18]

H. C. Fattorini and S. S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proceedings of the Royal Society of London Series A, 124 (1994), 211-251.  doi: 10.1017/S0308210500028444.  Google Scholar

[19] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, 1999.  doi: 10.1017/CBO9780511574795.  Google Scholar
[20]

H. O. Fattorini and S. S. Sritharan, Optimal control problems with state constraints in fluid mechanics and combustion, Applied Mathematics and Optimization, 38 (1998), 159-192.  doi: 10.1007/s002459900087.  Google Scholar

[21]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier–Stokes equations, Journal of Mathematical Fluid Mechanics, 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[22]

A. V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications, American Mathemtical Society, Rhode Island, 2000. doi: 10.1090/mmono/187.  Google Scholar

[23]

G. Geymonat and P. Leyland, Transport and propagation of a perturbation of a flow of a compressible fluid in a bounded region, Archive for Rational Mechanics and Analysis, 100 (1987), 53-81.  doi: 10.1007/BF00281247.  Google Scholar

[24]

M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM's Advances in Design and Control series, Philadelphia, 2003.  Google Scholar

[25]

A. JamesonN. Pierce and L. Martinelli, Optimum aerodynamic design using the Navier–Stokes equations, Theoretical Computational Fluid Dynamics, 10 (1998), 213-237.  doi: 10.2514/6.1997-101.  Google Scholar

[26]

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

[27]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, 1971.  Google Scholar

[28] P. L. Lions, Mathematical Topics in Fluid Mechanics, Volume 2: Compressible Models, Clarendon Press, 1998.   Google Scholar
[29]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[30]

D. MitraM. Ramaswamy and J.-P. Raymond, Local stabilization of compressible navier-stokes equations in one dimension around non-zero velocity, Advances in Differential Equations, 22 (2017), 693-736.   Google Scholar

[31]

J. Neustupa, A semigroup generated by the linearized Navier-Stokes equations for compressible fluid and its uniform growth bound in Hölder spaces, in Navier-Stokes Equations: Theory and Numerical Methods (ed. R. Salvi), Pitman. Research Notes Math. Ser., (1998), 86–100.  Google Scholar

[32]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and Its Applications, OUP Oxford, 2004.  Google Scholar

[33]

J. J. Otero, A. S. Sharma and R. D. Sandberg, Adjoint-based optimal flow control for compressible DNS, preprint, arXiv: 1603.05887v2. Google Scholar

[34]

J.-P. Penot, Calculus Without Derivatives, Springer, 2013. doi: 10.1007/978-1-4614-4538-8.  Google Scholar

[35]

V. A. Solonnikov, Solvability of the initial–boundary-value problem for the equations of motion of a viscous compressible fluid, Zap. Nauchn. Semin. LOMI, 56, (1976), 128–142.  Google Scholar

[36]

V. A. Solonnikov, On the solvability of initial-boundary value problems for a viscous compressible fluid in an infinite time interval, St. Petersburg Math. J., 27 (2016), 523-546.  doi: 10.1090/spmj/1402.  Google Scholar

[37]

S. S. Sritharan, Optimal Control of Viscous Flow, SIAM Frontiers in Applied Mathematics, Philadelphia, 1998. doi: 10.1137/1.9781611971415.  Google Scholar

[38]

G. Ströhmer, About compressible viscous fluid flow in a bounded region, Pacific J. of Math., 143 (1990), 359-375.  doi: 10.2140/pjm.1990.143.359.  Google Scholar

[39]

G. Ströhmer, About the resolvent of an operator from fluid dynamics, Math. Z., 194 (1987), 183-191.  doi: 10.1007/BF01161967.  Google Scholar

[40]

A. Valli, Periodic and stationary solutions for compressible Navier–Stokes equations via a stability method, Annali della Scuola Normale Superiore di Pisa–Classe di Scienze, 10 (1983), 607–647.  Google Scholar

[41]

G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Analysis, 52 (2003), 1911-1931.  doi: 10.1016/S0362-546X(02)00282-1.  Google Scholar

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer-Verlag, 2006.  Google Scholar

[2]

E. V. Amosova, Optimal control of a viscous heat-conducting gas flow, Journal of Applied and Industrial Mathematics, 3 (2009), 5-20.  doi: 10.1134/S1990478909010025.  Google Scholar

[3]

V. Barbu, Optimal Control of Variational Inequalities, Pitman Advanced Publishing Program, 1984.  Google Scholar

[4]

V. Barbu, Mathematical Methods in Optimization in Differential Systems, Springer, 1994. doi: 10.1007/978-94-011-0760-0.  Google Scholar

[5]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar

[6]

P. BellaE. FeireislB. J. Jin and A. Novotný, Robustness of strong solutions to the compressible Navier–Stokes system, Mathematische Annalen, 362 (2015), 281-303.  doi: 10.1007/s00208-014-1119-2.  Google Scholar

[7]

J. Borggaard and J. Burns, A PDE sensitivity equation method for optimal aerodynamic design, Journal of Computational Physics, 136 (1997), 366-384.  doi: 10.1006/jcph.1997.5743.  Google Scholar

[8]

Y. Cho and H. Kim, On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities, Manuscripta Mathematica, 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[9]

S. Chowdhury and M. Ramaswamy, Optimal control of linearized compressible Navier–Stokes equations, ESAIM: Control, Optimization and Calculus of Variations, 19 (2013), 587-615.  doi: 10.1051/cocv/2012023.  Google Scholar

[10]

S. ChowdhuryM. Ramaswamy and J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier–Stokes system in one-dimension, SIAM Journal on Control and Optimization, 50 (2012), 2959-2987.  doi: 10.1137/110846683.  Google Scholar

[11]

S. S. Collis, K. Ghayour, M. Heinkenschloss, M. Ulbrich and S. Ulbrich, Towards adjoint-based methods for aeroacoustic control, in 39th Aerospace Science Meeting & Exhibit, Reno, NV, AIAA PAPER, (2001), 2001-821, 1–17. doi: 10.2514/6.2001-821.  Google Scholar

[12]

S. S. Collis, K. Ghayour, M. Heinkenschloss, M. Ulbrich and S. Ulbrich, Numerical solution of optimal control problems governed by the compressible Navier–Stokes equations, in Optimal Control of Complex Structures: International Conference in Oberwolfach, Birkhäuser Basel, Basel, (2002), 43–55.  Google Scholar

[13]

H. B. da Veiga, Diffusion on viscous fluids, Existence and asymptotic properties of solutions, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, 10 (1983), 341–355.  Google Scholar

[14]

S. DoboszczakM. T. Mohan and S. S. Sritharan, Existence of optimal controls for compressible viscous flow, Journal of Mathematical Fluid Mechanics, 20 (2018), 199-211.  doi: 10.1007/s00021-017-0318-5.  Google Scholar

[15]

I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications, 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

I. Ekeland, Nonconvex minimization problems, Bulletin of the American Mathematical Society, 1 (1979), 443-473.  doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar

[17]

S. ErvedozaO. GlassS. Guerrero and J.-P. Puel, Local exact controllability for the one–dimensional compressible Navier–Stokes equation, Arch. Rational Mech. Anal., 206 (2012), 189-238.  doi: 10.1007/s00205-012-0534-3.  Google Scholar

[18]

H. C. Fattorini and S. S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proceedings of the Royal Society of London Series A, 124 (1994), 211-251.  doi: 10.1017/S0308210500028444.  Google Scholar

[19] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, 1999.  doi: 10.1017/CBO9780511574795.  Google Scholar
[20]

H. O. Fattorini and S. S. Sritharan, Optimal control problems with state constraints in fluid mechanics and combustion, Applied Mathematics and Optimization, 38 (1998), 159-192.  doi: 10.1007/s002459900087.  Google Scholar

[21]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier–Stokes equations, Journal of Mathematical Fluid Mechanics, 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[22]

A. V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications, American Mathemtical Society, Rhode Island, 2000. doi: 10.1090/mmono/187.  Google Scholar

[23]

G. Geymonat and P. Leyland, Transport and propagation of a perturbation of a flow of a compressible fluid in a bounded region, Archive for Rational Mechanics and Analysis, 100 (1987), 53-81.  doi: 10.1007/BF00281247.  Google Scholar

[24]

M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM's Advances in Design and Control series, Philadelphia, 2003.  Google Scholar

[25]

A. JamesonN. Pierce and L. Martinelli, Optimum aerodynamic design using the Navier–Stokes equations, Theoretical Computational Fluid Dynamics, 10 (1998), 213-237.  doi: 10.2514/6.1997-101.  Google Scholar

[26]

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

[27]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, 1971.  Google Scholar

[28] P. L. Lions, Mathematical Topics in Fluid Mechanics, Volume 2: Compressible Models, Clarendon Press, 1998.   Google Scholar
[29]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[30]

D. MitraM. Ramaswamy and J.-P. Raymond, Local stabilization of compressible navier-stokes equations in one dimension around non-zero velocity, Advances in Differential Equations, 22 (2017), 693-736.   Google Scholar

[31]

J. Neustupa, A semigroup generated by the linearized Navier-Stokes equations for compressible fluid and its uniform growth bound in Hölder spaces, in Navier-Stokes Equations: Theory and Numerical Methods (ed. R. Salvi), Pitman. Research Notes Math. Ser., (1998), 86–100.  Google Scholar

[32]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and Its Applications, OUP Oxford, 2004.  Google Scholar

[33]

J. J. Otero, A. S. Sharma and R. D. Sandberg, Adjoint-based optimal flow control for compressible DNS, preprint, arXiv: 1603.05887v2. Google Scholar

[34]

J.-P. Penot, Calculus Without Derivatives, Springer, 2013. doi: 10.1007/978-1-4614-4538-8.  Google Scholar

[35]

V. A. Solonnikov, Solvability of the initial–boundary-value problem for the equations of motion of a viscous compressible fluid, Zap. Nauchn. Semin. LOMI, 56, (1976), 128–142.  Google Scholar

[36]

V. A. Solonnikov, On the solvability of initial-boundary value problems for a viscous compressible fluid in an infinite time interval, St. Petersburg Math. J., 27 (2016), 523-546.  doi: 10.1090/spmj/1402.  Google Scholar

[37]

S. S. Sritharan, Optimal Control of Viscous Flow, SIAM Frontiers in Applied Mathematics, Philadelphia, 1998. doi: 10.1137/1.9781611971415.  Google Scholar

[38]

G. Ströhmer, About compressible viscous fluid flow in a bounded region, Pacific J. of Math., 143 (1990), 359-375.  doi: 10.2140/pjm.1990.143.359.  Google Scholar

[39]

G. Ströhmer, About the resolvent of an operator from fluid dynamics, Math. Z., 194 (1987), 183-191.  doi: 10.1007/BF01161967.  Google Scholar

[40]

A. Valli, Periodic and stationary solutions for compressible Navier–Stokes equations via a stability method, Annali della Scuola Normale Superiore di Pisa–Classe di Scienze, 10 (1983), 607–647.  Google Scholar

[41]

G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Analysis, 52 (2003), 1911-1931.  doi: 10.1016/S0362-546X(02)00282-1.  Google Scholar

[1]

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 & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[2]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[3]

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

[4]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[5]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499

[6]

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 & Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[7]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[8]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[9]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

[10]

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

[11]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[12]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[13]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[14]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021031

[15]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[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 & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

[17]

Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291

[18]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

[19]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[20]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (94)
  • HTML views (304)
  • Cited by (0)

[Back to Top]