# American Institute of Mathematical Sciences

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  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 & Related Fields, 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.  Google Scholar [2] P. Agarwal, U. 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.  Google Scholar [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.  Google Scholar [4] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, vol. 190, Academic Press, Inc., Boston, MA, 1993.  Google Scholar [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.  Google Scholar [6] T. Biswas, S. 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.  Google Scholar [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.  Google Scholar [8] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, PA, 2013.  Google Scholar [9] S. Doboszczak, M. 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.  Google Scholar [10] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar [11] I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, University of Chicago Press, Chicago, IL, 1983.  Google Scholar [12] A. V. Fursikov, Optimal control of distributed systems: Theory and applications, American Mathematical Society, Providence, RI, (2000). doi: 10.1090/mmono/187.  Google Scholar [13] G. Galilei, Dialogue Concerning the Two Chief World Systems, 1632. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. A. Kagan, Hydrodynamic Models of Tidal Motions in the Sea, Gidrometeoizdat, Leningrad, 1968. Google Scholar [19] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar [20] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.  Google Scholar [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.  Google Scholar [22] G. I. Marchuk and B. A. Kagan, Ocean Tides: Mathematical Models and Numerical Experiments, Pergamon Press, Elmsford, NY, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] M. T. Mohan, Necessary conditions for distributed optimal control of two dimensional tidal dynamics system with state constraints, work-in-progress, (2020). Google Scholar [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.   Google Scholar [29] I. Newton, Philosophiae Naturalis Principia Mathematica, William Dawson & Sons, Ltd., London, 1687.  Google Scholar [30] J. Pedlosky, Geophysical Fluid Dyanmics I, II, Springer, Heidelberg, 1981. Google Scholar [31] J. P. Raymond, Optimal control of partial differential equations, Université Paul Sabatier, Lecture Notes, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.  Google Scholar [36] S. S. Sritharan, Optimal Control of Viscous Flow, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971415.  Google Scholar [37] M. Suvinthra, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] H. Yin, Stochastic analysis of backward tidal dynamics equation, Commun. Stoch. Anal., 5 (2011), 745-768.  doi: 10.31390/cosa.5.4.09.  Google Scholar

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.  Google Scholar [2] P. Agarwal, U. 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.  Google Scholar [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.  Google Scholar [4] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, vol. 190, Academic Press, Inc., Boston, MA, 1993.  Google Scholar [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.  Google Scholar [6] T. Biswas, S. 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.  Google Scholar [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.  Google Scholar [8] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, PA, 2013.  Google Scholar [9] S. Doboszczak, M. 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.  Google Scholar [10] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar [11] I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, University of Chicago Press, Chicago, IL, 1983.  Google Scholar [12] A. V. Fursikov, Optimal control of distributed systems: Theory and applications, American Mathematical Society, Providence, RI, (2000). doi: 10.1090/mmono/187.  Google Scholar [13] G. Galilei, Dialogue Concerning the Two Chief World Systems, 1632. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. A. Kagan, Hydrodynamic Models of Tidal Motions in the Sea, Gidrometeoizdat, Leningrad, 1968. Google Scholar [19] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar [20] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.  Google Scholar [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.  Google Scholar [22] G. I. Marchuk and B. A. Kagan, Ocean Tides: Mathematical Models and Numerical Experiments, Pergamon Press, Elmsford, NY, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] M. T. Mohan, Necessary conditions for distributed optimal control of two dimensional tidal dynamics system with state constraints, work-in-progress, (2020). Google Scholar [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.   Google Scholar [29] I. Newton, Philosophiae Naturalis Principia Mathematica, William Dawson & Sons, Ltd., London, 1687.  Google Scholar [30] J. Pedlosky, Geophysical Fluid Dyanmics I, II, Springer, Heidelberg, 1981. Google Scholar [31] J. P. Raymond, Optimal control of partial differential equations, Université Paul Sabatier, Lecture Notes, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.  Google Scholar [36] S. S. Sritharan, Optimal Control of Viscous Flow, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971415.  Google Scholar [37] M. Suvinthra, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] H. Yin, Stochastic analysis of backward tidal dynamics equation, Commun. Stoch. Anal., 5 (2011), 745-768.  doi: 10.31390/cosa.5.4.09.  Google Scholar
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