• Previous Article
    Optimal trend-following trading rules under a three-state regime switching model
  • MCRF Home
  • This Issue
  • Next Article
    Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms
March  2012, 2(1): 61-80. doi: 10.3934/mcrf.2012.2.61

Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China, China

Received  June 2011 Revised  November 2011 Published  January 2012

This paper deals with the Pontryagin's principle of optimal control problems governed by the 2D Navier-Stokes equations with integral state constraints and coupled integral control--state constraints. As an application, the necessary conditions for the local solution in the sense of $L^r(0,T;L^2(\Omega))$ ($2 < r < \infty$) are also obtained.
Citation: 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
References:
[1]

2nd edition, North-Holland, Amsterdam, 1979. Google Scholar

[2]

Dimensional System," Birkhäuser, Boston, 1995. Google Scholar

[3]

Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.  Google Scholar

[4]

Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.  Google Scholar

[5]

Nonlinear Anal., 73 (2010), 3924-3939. doi: 10.1016/j.na.2010.08.026.  Google Scholar

[6]

SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.  Google Scholar

[7]

SIAM J. Control Optim., 39 (2000), 1182-1203.  Google Scholar

[8]

SIAM J. Control Optim., 36 (1998), 1853-1879. Google Scholar

[9]

Math. Control Relat. Fields, 1 (2011), 267-306. doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[10]

Math. Control Relat. Fields, 1 (2011), 189-230. doi: 10.3934/mcrf.2011.1.189.  Google Scholar

[11]

Systems Control Lett., 30 (1997), 93-100. doi: 10.1016/S0167-6911(96)00083-7.  Google Scholar

[12]

Math. Control Relat. Fields, 1 (2011), 307-330. doi: 10.3934/mcrf.2011.1.307.  Google Scholar

[13]

J. Inv. Ill-Posed Problems, 12 (2004), 43-123; Part II: $L_2(\Omega)$-estimates, J. Inv. Ill-Posed Problems, 12 (2004), 183-231, (MR2061430).  Google Scholar

[14]

Math. Control Relat. Fields, 1 (2011), 149-175. doi: 10.3934/mcrf.2011.1.149.  Google Scholar

[15]

Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177.  Google Scholar

[16]

CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983. Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211-251. doi: 10.1017/S0308210500028444.  Google Scholar

[18]

in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.  Google Scholar

[19]

Springer-Verlag, New York, 1983. Google Scholar

[20]

Nonlinear Anal., 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.  Google Scholar

[21]

SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.  Google Scholar

show all references

References:
[1]

2nd edition, North-Holland, Amsterdam, 1979. Google Scholar

[2]

Dimensional System," Birkhäuser, Boston, 1995. Google Scholar

[3]

Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.  Google Scholar

[4]

Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.  Google Scholar

[5]

Nonlinear Anal., 73 (2010), 3924-3939. doi: 10.1016/j.na.2010.08.026.  Google Scholar

[6]

SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.  Google Scholar

[7]

SIAM J. Control Optim., 39 (2000), 1182-1203.  Google Scholar

[8]

SIAM J. Control Optim., 36 (1998), 1853-1879. Google Scholar

[9]

Math. Control Relat. Fields, 1 (2011), 267-306. doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[10]

Math. Control Relat. Fields, 1 (2011), 189-230. doi: 10.3934/mcrf.2011.1.189.  Google Scholar

[11]

Systems Control Lett., 30 (1997), 93-100. doi: 10.1016/S0167-6911(96)00083-7.  Google Scholar

[12]

Math. Control Relat. Fields, 1 (2011), 307-330. doi: 10.3934/mcrf.2011.1.307.  Google Scholar

[13]

J. Inv. Ill-Posed Problems, 12 (2004), 43-123; Part II: $L_2(\Omega)$-estimates, J. Inv. Ill-Posed Problems, 12 (2004), 183-231, (MR2061430).  Google Scholar

[14]

Math. Control Relat. Fields, 1 (2011), 149-175. doi: 10.3934/mcrf.2011.1.149.  Google Scholar

[15]

Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177.  Google Scholar

[16]

CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983. Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211-251. doi: 10.1017/S0308210500028444.  Google Scholar

[18]

in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.  Google Scholar

[19]

Springer-Verlag, New York, 1983. Google Scholar

[20]

Nonlinear Anal., 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.  Google Scholar

[21]

SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.  Google Scholar

[1]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[2]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408

[3]

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

[4]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041

[5]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[6]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[7]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[8]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[9]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[10]

Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021059

[11]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007

[12]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215

[13]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[14]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[15]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[16]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040

[17]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026

[18]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[19]

Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021022

[20]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]