December  2021, 11(4): 739-769. doi: 10.3934/mcrf.2020045

First order necessary conditions of optimality for the two dimensional tidal dynamics system

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

*Corresponding author

Received  September 2019 Revised  August 2020 Published  December 2021 Early access  November 2020

Fund Project: M. T. Mohan is supported by INSPIRE Faculty Award-IFA17-MA110

In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems are established and the optimal control is characterized via the adjoint variable. We also establish the uniqueness of optimal control in small time interval.

Citation: Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control and Related Fields, 2021, 11 (4) : 739-769. doi: 10.3934/mcrf.2020045
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325.  doi: 10.1007/BF00271794.

[2]

P. AgarwalU. Manna and D. Mukherjee, Stochastic control of tidal dynamics equation with Lévy noise, Appl. Math. Optim., 79 (2019), 327-396.  doi: 10.1007/s00245-017-9440-2.

[3]

V. I. Agoshkov and E. A. Botvinovsky, Numerical solution of a hyperbolic-parabolic system by splitting methods and optimal control approaches, Comput. Methods Appl. Math., 7 (2007), 193-207.  doi: 10.2478/cmam-2007-0011.

[4]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, vol. 190, Academic Press, Inc., Boston, MA, 1993.

[5]

N. R. C. Birkett and N. K. Nichols, Optimal control problems in tidal power generation, Industrial Numerical Analysis, Oxford Sci. Publ., Oxford Univ. Press, New York, 1986, 53-89.

[6]

T. BiswasS. Dharmatti and M. T. Mohan, Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, Analysis (Berlin), 40 (2020), 127-150.  doi: 10.1515/anly-2019-0049.

[7]

T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, J. Math. Fluid Mech., 22 (2020), Art. 34, 42 pp. doi: 10.1007/s00021-020-00493-8.

[8]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, PA, 2013.

[9]

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

[10]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.

[11]

I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, University of Chicago Press, Chicago, IL, 1983.

[12]

A. V. Fursikov, Optimal control of distributed systems: Theory and applications, American Mathematical Society, Providence, RI, (2000). doi: 10.1090/mmono/187.

[13]

G. Galilei, Dialogue Concerning the Two Chief World Systems, 1632.

[14]

R. G. Gordeev, The existence of a periodic solution in tide dynamic problem, Journal of Soviet Mathematics, 6 (1976), 1-4.  doi: 10.1007/BF01084856.

[15]

M. D. Gunzburger, Perspectives in Flow Control and Optimization, Advances in Design and Control, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.

[16]

A. Haseena, M. Suvinthra, M. T. Mohan and K. Balachandran, Moderate deviations for stochastic tidal dynamics equation with multiplicative noise, Applicable Analysis, 2020. doi: 10.1080/00036811.2020.1781827.

[17]

V. M. Ipatova, Solvability of a tide dynamics model in adjacent seas, Russian J. Numer. Anal. Math. Modelling, 20 (2005), 67-79.  doi: 10.1515/1569398053270822.

[18]

B. A. Kagan, Hydrodynamic Models of Tidal Motions in the Sea, Gidrometeoizdat, Leningrad, 1968.

[19]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[20]

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

[21]

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

[22]

G. I. Marchuk and B. A. Kagan, Ocean Tides: Mathematical Models and Numerical Experiments, Pergamon Press, Elmsford, NY, 1984.

[23]

G. I. Marchuk and B. A. Kagan, Dynamics of Ocean Tides, Kluwer Academic Publishers, Dordrecht/Boston/London, 1989. doi: 10.1007/978-94-009-2571-7.

[24]

U. Manna, J. L. Menaldi and S. S. Sritharan, Stochastic analysis of tidal dynamics equation, Infinite Dimensional Stochastic Analysis, World Sci. Publ., Hackensack, NJ, (2008), 90–113. doi: 10.1142/9789812779557_0006.

[25]

M. T. Mohan, On the two dimensional tidal dynamics system: Stationary solution and stability, Appl. Anal., 99 (2020), 1795-1826.  doi: 10.1080/00036811.2018.1546002.

[26]

M. T. Mohan, Dynamic programming and feedback analysis of the two dimensional tidal dynamics system, in ESAIM: Control, Optimisation and Calculus of Variations, 2020. doi: 10.1051/cocv/2020025.

[27]

M. T. Mohan, Necessary conditions for distributed optimal control of two dimensional tidal dynamics system with state constraints, work-in-progress, (2020).

[28]

R. Mosetti, Optimal control of sea level in a tidal basin by means of the Pontryagin maximum principle, Applied Mathematical Modelling, 9 (1985), 321-324. 

[29]

I. Newton, Philosophiae Naturalis Principia Mathematica, William Dawson & Sons, Ltd., London, 1687.

[30]

J. Pedlosky, Geophysical Fluid Dyanmics I, II, Springer, Heidelberg, 1981.

[31]

J. P. Raymond, Optimal control of partial differential equations, Université Paul Sabatier, Lecture Notes, 2013.

[32]

S. C. Ryrie and D. T. Bickley, Optimally controlled hydrodynamics for tidal power in the Severn Estuary, Appl. Math. Modelling, 9 (1985), 1-10.  doi: 10.1016/0307-904X(85)90134-9.

[33]

S. C. Ryrie, An optimal control model of tidal power generation, Appl. Math. Modelling, 19 (1985), 123-126.  doi: 10.1016/0307-904X(94)00012-U.

[34]

J. Simon, Compact sets in the space $\mathrm{L}^p(0, T;\mathrm{B})$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[35]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

[36]

S. S. Sritharan, Optimal Control of Viscous Flow, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971415.

[37]

M. SuvinthraS. S. Sritharan and K. Balachandran, Large deviations for stochastic tidal dynamics equation, Commun. Stoch. Anal., 9 (2015), 477-502.  doi: 10.31390/cosa.9.4.04.

[38]

H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  doi: 10.1090/S0002-9947-1934-1501735-3.

[39]

Z. Yanga and J. M. Hamrickb, Optimal control of salinity boundary condition in a tidal model using a variational inverse method, Estuarine, Coastal and Shelf Science, 62 (2005), 13-24.  doi: 10.1016/j.ecss.2004.08.003.

[40]

H. Yin, Stochastic analysis of backward tidal dynamics equation, Commun. Stoch. Anal., 5 (2011), 745-768.  doi: 10.31390/cosa.5.4.09.

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325.  doi: 10.1007/BF00271794.

[2]

P. AgarwalU. Manna and D. Mukherjee, Stochastic control of tidal dynamics equation with Lévy noise, Appl. Math. Optim., 79 (2019), 327-396.  doi: 10.1007/s00245-017-9440-2.

[3]

V. I. Agoshkov and E. A. Botvinovsky, Numerical solution of a hyperbolic-parabolic system by splitting methods and optimal control approaches, Comput. Methods Appl. Math., 7 (2007), 193-207.  doi: 10.2478/cmam-2007-0011.

[4]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, vol. 190, Academic Press, Inc., Boston, MA, 1993.

[5]

N. R. C. Birkett and N. K. Nichols, Optimal control problems in tidal power generation, Industrial Numerical Analysis, Oxford Sci. Publ., Oxford Univ. Press, New York, 1986, 53-89.

[6]

T. BiswasS. Dharmatti and M. T. Mohan, Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, Analysis (Berlin), 40 (2020), 127-150.  doi: 10.1515/anly-2019-0049.

[7]

T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, J. Math. Fluid Mech., 22 (2020), Art. 34, 42 pp. doi: 10.1007/s00021-020-00493-8.

[8]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, PA, 2013.

[9]

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

[10]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.

[11]

I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, University of Chicago Press, Chicago, IL, 1983.

[12]

A. V. Fursikov, Optimal control of distributed systems: Theory and applications, American Mathematical Society, Providence, RI, (2000). doi: 10.1090/mmono/187.

[13]

G. Galilei, Dialogue Concerning the Two Chief World Systems, 1632.

[14]

R. G. Gordeev, The existence of a periodic solution in tide dynamic problem, Journal of Soviet Mathematics, 6 (1976), 1-4.  doi: 10.1007/BF01084856.

[15]

M. D. Gunzburger, Perspectives in Flow Control and Optimization, Advances in Design and Control, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.

[16]

A. Haseena, M. Suvinthra, M. T. Mohan and K. Balachandran, Moderate deviations for stochastic tidal dynamics equation with multiplicative noise, Applicable Analysis, 2020. doi: 10.1080/00036811.2020.1781827.

[17]

V. M. Ipatova, Solvability of a tide dynamics model in adjacent seas, Russian J. Numer. Anal. Math. Modelling, 20 (2005), 67-79.  doi: 10.1515/1569398053270822.

[18]

B. A. Kagan, Hydrodynamic Models of Tidal Motions in the Sea, Gidrometeoizdat, Leningrad, 1968.

[19]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[20]

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

[21]

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

[22]

G. I. Marchuk and B. A. Kagan, Ocean Tides: Mathematical Models and Numerical Experiments, Pergamon Press, Elmsford, NY, 1984.

[23]

G. I. Marchuk and B. A. Kagan, Dynamics of Ocean Tides, Kluwer Academic Publishers, Dordrecht/Boston/London, 1989. doi: 10.1007/978-94-009-2571-7.

[24]

U. Manna, J. L. Menaldi and S. S. Sritharan, Stochastic analysis of tidal dynamics equation, Infinite Dimensional Stochastic Analysis, World Sci. Publ., Hackensack, NJ, (2008), 90–113. doi: 10.1142/9789812779557_0006.

[25]

M. T. Mohan, On the two dimensional tidal dynamics system: Stationary solution and stability, Appl. Anal., 99 (2020), 1795-1826.  doi: 10.1080/00036811.2018.1546002.

[26]

M. T. Mohan, Dynamic programming and feedback analysis of the two dimensional tidal dynamics system, in ESAIM: Control, Optimisation and Calculus of Variations, 2020. doi: 10.1051/cocv/2020025.

[27]

M. T. Mohan, Necessary conditions for distributed optimal control of two dimensional tidal dynamics system with state constraints, work-in-progress, (2020).

[28]

R. Mosetti, Optimal control of sea level in a tidal basin by means of the Pontryagin maximum principle, Applied Mathematical Modelling, 9 (1985), 321-324. 

[29]

I. Newton, Philosophiae Naturalis Principia Mathematica, William Dawson & Sons, Ltd., London, 1687.

[30]

J. Pedlosky, Geophysical Fluid Dyanmics I, II, Springer, Heidelberg, 1981.

[31]

J. P. Raymond, Optimal control of partial differential equations, Université Paul Sabatier, Lecture Notes, 2013.

[32]

S. C. Ryrie and D. T. Bickley, Optimally controlled hydrodynamics for tidal power in the Severn Estuary, Appl. Math. Modelling, 9 (1985), 1-10.  doi: 10.1016/0307-904X(85)90134-9.

[33]

S. C. Ryrie, An optimal control model of tidal power generation, Appl. Math. Modelling, 19 (1985), 123-126.  doi: 10.1016/0307-904X(94)00012-U.

[34]

J. Simon, Compact sets in the space $\mathrm{L}^p(0, T;\mathrm{B})$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[35]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

[36]

S. S. Sritharan, Optimal Control of Viscous Flow, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971415.

[37]

M. SuvinthraS. S. Sritharan and K. Balachandran, Large deviations for stochastic tidal dynamics equation, Commun. Stoch. Anal., 9 (2015), 477-502.  doi: 10.31390/cosa.9.4.04.

[38]

H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  doi: 10.1090/S0002-9947-1934-1501735-3.

[39]

Z. Yanga and J. M. Hamrickb, Optimal control of salinity boundary condition in a tidal model using a variational inverse method, Estuarine, Coastal and Shelf Science, 62 (2005), 13-24.  doi: 10.1016/j.ecss.2004.08.003.

[40]

H. Yin, Stochastic analysis of backward tidal dynamics equation, Commun. Stoch. Anal., 5 (2011), 745-768.  doi: 10.31390/cosa.5.4.09.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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]

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

[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]

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

[11]

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 and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[12]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485

[13]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control and Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

[14]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445

[15]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[16]

Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021059

[17]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.284

Article outline

[Back to Top]