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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 and 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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

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