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April  2022, 11(2): 347-371. doi: 10.3934/eect.2020110

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

Stefan Doboszczak 1,, , Manil T. Mohan 2, and  Sivaguru S. Sritharan 3,

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 Published  April 2022 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.

Keywords: Compressible Navier-Stokes, optimal control, Pontryagin maximum principle, necessary conditions, Ekeland variational principle, state constraint.
Mathematics Subject Classification: Primary: 49K20, 76N25; Secondary: 49J20, 35Q30.
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 and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110
References:
[1]

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

[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.

[3]

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

[4]

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

[5]

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

[6]

P. Bella, E. Feireisl, B. 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.

[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.

[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.

[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.

[10]

S. Chowdhury, M. 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.

[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.

[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.

[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.

[14]

S. Doboszczak, M. 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.

[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.

[16]

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

[17]

S. Ervedoza, O. Glass, S. 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.

[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.

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

[21]

E. Feireisl, A. 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.

[22]

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

[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.

[24]

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

[25]

A. Jameson, N. 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.

[26]

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

[27]

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

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

[30]

D. Mitra, M. 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. 

[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.

[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.

[33]

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

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[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.

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[5]

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

[6]

P. Bella, E. Feireisl, B. 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.

[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.

[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.

[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.

[10]

S. Chowdhury, M. 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.

[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.

[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.

[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.

[14]

S. Doboszczak, M. 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.

[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.

[16]

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

[17]

S. Ervedoza, O. Glass, S. 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.

[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.

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

[21]

E. Feireisl, A. 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.

[22]

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

[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.

[24]

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

[25]

A. Jameson, N. 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.

[26]

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

[27]

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

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

[30]

D. Mitra, M. 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. 

[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.

[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.

[33]

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

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[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.

[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.

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