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March  2021, 10(1): 37-59. doi: 10.3934/eect.2020051

Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions

Elimhan N. Mahmudov ,

Department of Mathematics, Istanbul Technical University, Istanbul, Turkey, Azerbaijan National Academy of Sciences Institute of Control Systems, Baku, Azerbaijan

* Corresponding author: elimhan22@yahoo.com

Received  May 2019 Revised  January 2020 Published  March 2021 Early access  May 2020

The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. At the end of the paper duality in problems with second order linear discrete and continuous models and model of control problem with polyhedral DFIs are considered.

Keywords: Infimal convolution, duality, conjugate, Euler-Lagrange, polyhedral, discrete-approximate.
Mathematics Subject Classification: 34A60; 49J15; 49K25; 90C46.
Citation: Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051
References:
[1]

S. Artstein-Avidan and V. Milman, A characterization of the concept of duality, Electron. Res. Announc. Math. Sci., 14 (2007), 42-59.   Google Scholar

[2]

D. Azzam-Laouir and F. Selamnia, On state-dependent sweeping process in Banach spaces, Evol. Equ. Control Theory, 7 (2018), 183-196.  doi: 10.3934/eect.2018009.  Google Scholar

[3]

V. Barbu, I. Lasiecka, D. Tiba and C. Varsan, Analysis and optimization of differential systems, IFIP TC7/WG7.2 International Working Conference Held in Constanta, September 10-14, 2002 doi: 10.1007/978-0-387-35690-7.  Google Scholar

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S. A. Belbas and S. M. Lenhart, Deterministic optimal control problem with final state constraints, The 23rd IEEE Conference On Decision and Control, (1984), 526–527. doi: 10.1109/CDC.1984.272051.  Google Scholar

[5]

A. Bressan, Differential inclusions and the control of forest fires, J. Differential Equations, 243 (2007), 179-207.  doi: 10.1016/j.jde.2007.03.009.  Google Scholar

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G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Rev. Mat. Complut., 24 (2011), 83-94.  doi: 10.1007/s13163-010-0030-y.  Google Scholar

[7]

P. Cannarsa, A. Marigonda and K. T. Nguyen, Optimality conditions and regularity results for time optimal control problems with differential inclusions, J. Math. Anal. Appl., 427 (2015), 202-228.  doi: 10.1016/j.jmaa.2015.02.027.  Google Scholar

[8]

A. Dhara and A. Mehra, Conjugate duality for generalized convex optimization problems, J. Ind. Manag. Optim., 3 (2007), 415-427.  doi: 10.3934/jimo.2007.3.415.  Google Scholar

[9]

M. D. Fajardol and J. Vidal, Necessary and sufficient conditions for strong Fenchel-Lagrange duality via a coupling conjugation scheme, J. Optim. Theory Appl., 176 (2018), 57-73.  doi: 10.1007/s10957-017-1209-x.  Google Scholar

[10]

A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: The three-dimensional case, SIAM J. Control Optim., 43 (2005), 2191-2232.  doi: 10.1137/S0363012904400805.  Google Scholar

[11]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, Series in Nonlinear Analysis and its Applications, Izdat "Nauka", Moscow, 1974.  Google Scholar

[12]

N. C. Kourogenis, Strongly nonlinear second order differential inclusions with generalized boundary conditions, J. Math. Anal. Appl., 287 (2003), 348-364.  doi: 10.1016/S0022-247X(02)00511-5.  Google Scholar

[13]

I. Lasiecka and N. Fourrier, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631.  Google Scholar

[14]

P.-J. Laurent, Approximation et optimisation, in Collection Enseignment des Sciences, 13, Herman, Paris, 1972.  Google Scholar

[15]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Inc., Amsterdam, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar

[16]

E. N. Mahmudov, On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type, J. Math. Anal. Appl., 307 (2005), 628-640.  doi: 10.1016/j.jmaa.2005.01.037.  Google Scholar

[17]

E. N. Mahmudov and M. E. Unal, Optimal control of discrete and differential inclusions with distributed parameters in the gradient form, J. Dyn. Control Syst., 18 (2012), 83-101.  doi: 10.1007/s10883-012-9135-6.  Google Scholar

[18]

E. N. Mahmudov, Optimization of fourth-order differential inclusions, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44 (2018), 90-106.   Google Scholar

[19]

E. N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville-type differential inclusions, J. Optim. Theory Appl., 177 (2018), 345-375.  doi: 10.1007/s10957-018-1260-2.  Google Scholar

[20]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Ind. Manag. Optim., 16 (2020), 169-187.  doi: 10.3934/jimo.2018145.  Google Scholar

[21]

E. N. Mahmudov, Optimal control of second Order delay-discrete and delay-differential inclusions with state constraints, Evol. Equ. Control Theory, 7 (2018), 501-529.  doi: 10.3934/eect.2018024.  Google Scholar

[22]

E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: Control, Optimisation and Calculus of Variations, (2019). doi: 10.1051/cocv/2019018.  Google Scholar

[23]

B. S. Mordukhovich and T. H. Cao, Optimal control of a nonconvex perturbed sweeping process, J. Differential Equations, 266 (2019), 1003-1050.  doi: 10.1016/j.jde.2018.07.066.  Google Scholar

[24]

N. S. Papageorgiou and V. D. Rădulescu, Periodic solutions for time-dependent subdifferential evolution inclusions, Evol. Equ. Control Theory, 6 (2017), 277-297.  doi: 10.3934/eect.2017015.  Google Scholar

[25]

R. T. Rockafellar and P. R. Wolenski, Convexity in Hamilton-Jacobi theory. 1. Dynamics and duality, SIAM J. Control Optim., 39 (2000), 1323-1350.  doi: 10.1137/S0363012998345366.  Google Scholar

[26]

T. I. Seidman, Compactness of a fixpoint set and optimal control, Appl. Anal., 88 (2009), 419-423.  doi: 10.1080/00036810902766708.  Google Scholar

[27]

S. Sharma, A. Jayswal and S. Choudhury, Sufficiency and mixed type duality for multiobjective variational control problems involving $\alpha$-V-univexity, Evol. Equ. Control Theory, 6 (2017), 93-109.  doi: 10.3934/eect.2017006.  Google Scholar

[28]

Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.  Google Scholar

[29]

Q. Zhang and G. Li, Nonlinear boundary value problems for second order differential inclusions, Nonlinear Anal., 70 (2009), 3390-3406.  doi: 10.1016/j.na.2008.05.007.  Google Scholar

show all references

References:
[1]

S. Artstein-Avidan and V. Milman, A characterization of the concept of duality, Electron. Res. Announc. Math. Sci., 14 (2007), 42-59.   Google Scholar

[2]

D. Azzam-Laouir and F. Selamnia, On state-dependent sweeping process in Banach spaces, Evol. Equ. Control Theory, 7 (2018), 183-196.  doi: 10.3934/eect.2018009.  Google Scholar

[3]

V. Barbu, I. Lasiecka, D. Tiba and C. Varsan, Analysis and optimization of differential systems, IFIP TC7/WG7.2 International Working Conference Held in Constanta, September 10-14, 2002 doi: 10.1007/978-0-387-35690-7.  Google Scholar

[4]

S. A. Belbas and S. M. Lenhart, Deterministic optimal control problem with final state constraints, The 23rd IEEE Conference On Decision and Control, (1984), 526–527. doi: 10.1109/CDC.1984.272051.  Google Scholar

[5]

A. Bressan, Differential inclusions and the control of forest fires, J. Differential Equations, 243 (2007), 179-207.  doi: 10.1016/j.jde.2007.03.009.  Google Scholar

[6]

G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Rev. Mat. Complut., 24 (2011), 83-94.  doi: 10.1007/s13163-010-0030-y.  Google Scholar

[7]

P. Cannarsa, A. Marigonda and K. T. Nguyen, Optimality conditions and regularity results for time optimal control problems with differential inclusions, J. Math. Anal. Appl., 427 (2015), 202-228.  doi: 10.1016/j.jmaa.2015.02.027.  Google Scholar

[8]

A. Dhara and A. Mehra, Conjugate duality for generalized convex optimization problems, J. Ind. Manag. Optim., 3 (2007), 415-427.  doi: 10.3934/jimo.2007.3.415.  Google Scholar

[9]

M. D. Fajardol and J. Vidal, Necessary and sufficient conditions for strong Fenchel-Lagrange duality via a coupling conjugation scheme, J. Optim. Theory Appl., 176 (2018), 57-73.  doi: 10.1007/s10957-017-1209-x.  Google Scholar

[10]

A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: The three-dimensional case, SIAM J. Control Optim., 43 (2005), 2191-2232.  doi: 10.1137/S0363012904400805.  Google Scholar

[11]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, Series in Nonlinear Analysis and its Applications, Izdat "Nauka", Moscow, 1974.  Google Scholar

[12]

N. C. Kourogenis, Strongly nonlinear second order differential inclusions with generalized boundary conditions, J. Math. Anal. Appl., 287 (2003), 348-364.  doi: 10.1016/S0022-247X(02)00511-5.  Google Scholar

[13]

I. Lasiecka and N. Fourrier, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631.  Google Scholar

[14]

P.-J. Laurent, Approximation et optimisation, in Collection Enseignment des Sciences, 13, Herman, Paris, 1972.  Google Scholar

[15]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Inc., Amsterdam, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar

[16]

E. N. Mahmudov, On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type, J. Math. Anal. Appl., 307 (2005), 628-640.  doi: 10.1016/j.jmaa.2005.01.037.  Google Scholar

[17]

E. N. Mahmudov and M. E. Unal, Optimal control of discrete and differential inclusions with distributed parameters in the gradient form, J. Dyn. Control Syst., 18 (2012), 83-101.  doi: 10.1007/s10883-012-9135-6.  Google Scholar

[18]

E. N. Mahmudov, Optimization of fourth-order differential inclusions, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44 (2018), 90-106.   Google Scholar

[19]

E. N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville-type differential inclusions, J. Optim. Theory Appl., 177 (2018), 345-375.  doi: 10.1007/s10957-018-1260-2.  Google Scholar

[20]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Ind. Manag. Optim., 16 (2020), 169-187.  doi: 10.3934/jimo.2018145.  Google Scholar

[21]

E. N. Mahmudov, Optimal control of second Order delay-discrete and delay-differential inclusions with state constraints, Evol. Equ. Control Theory, 7 (2018), 501-529.  doi: 10.3934/eect.2018024.  Google Scholar

[22]

E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: Control, Optimisation and Calculus of Variations, (2019). doi: 10.1051/cocv/2019018.  Google Scholar

[23]

B. S. Mordukhovich and T. H. Cao, Optimal control of a nonconvex perturbed sweeping process, J. Differential Equations, 266 (2019), 1003-1050.  doi: 10.1016/j.jde.2018.07.066.  Google Scholar

[24]

N. S. Papageorgiou and V. D. Rădulescu, Periodic solutions for time-dependent subdifferential evolution inclusions, Evol. Equ. Control Theory, 6 (2017), 277-297.  doi: 10.3934/eect.2017015.  Google Scholar

[25]

R. T. Rockafellar and P. R. Wolenski, Convexity in Hamilton-Jacobi theory. 1. Dynamics and duality, SIAM J. Control Optim., 39 (2000), 1323-1350.  doi: 10.1137/S0363012998345366.  Google Scholar

[26]

T. I. Seidman, Compactness of a fixpoint set and optimal control, Appl. Anal., 88 (2009), 419-423.  doi: 10.1080/00036810902766708.  Google Scholar

[27]

S. Sharma, A. Jayswal and S. Choudhury, Sufficiency and mixed type duality for multiobjective variational control problems involving $\alpha$-V-univexity, Evol. Equ. Control Theory, 6 (2017), 93-109.  doi: 10.3934/eect.2017006.  Google Scholar

[28]

Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.  Google Scholar

[29]

Q. Zhang and G. Li, Nonlinear boundary value problems for second order differential inclusions, Nonlinear Anal., 70 (2009), 3390-3406.  doi: 10.1016/j.na.2008.05.007.  Google Scholar

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