September  2019, 8(3): 603-619. doi: 10.3934/eect.2019028

Optimal control of evolution differential inclusions with polynomial linear differential operators

1. 

Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

2. 

Azerbaijan National Academy of Sciences Institute of Control Systems, Baku, Azerbaijan

* Corresponding author: elimhan22@yahoo.com

Received  July 2018 Revised  December 2018 Published  May 2019

In this paper we have introduced a new class of problems of optimal control theory with differential inclusions described by polynomial linear differential operators. Consequently, there arises a rather complicated problem with simultaneous determination of the polynomial linear differential operators with variable coefficients and a Mayer functional depending on high order derivatives. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and transversality conditions are derived. Formulation of the transversality conditions at the endpoints of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with polynomial linear differential operators is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem and the Weierstrass-Pontryagin maximum principle is obtained. Then the optimality conditions are derived for second order convex differential inclusions with convex endpoint constraints.

Citation: Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
References:
[1]

A. Auslender and J. Mechler, Second order viability problems for differential inclusions, J. Math. Anal. Appl., 181 (1994), 205-218. doi: 10.1006/jmaa.1994.1015. Google Scholar

[2]

D. Azzam-LaouirC. Castaing and L. Thibault, Three boundary value problems for second order differential inclusion in Banach spaces, Contr. Cybernet., 31 (2002), 659-693. Google Scholar

[3]

V. Barbu and T. Precupanu, Convex control problems in banach spaces, Convexity and Optimization in Banach Spaces, (2012), 233-364. doi: 10.1007/978-94-007-2247-7_4. Google Scholar

[4]

V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal controls, Proceed.Steklov Inst.Mathem., 169 (1986), 199-259. Google Scholar

[5]

D. Bors and M. Majewski, On Mayer problem for systems governed by second-order ODE, Optimization, 63 (2014), 239-254. doi: 10.1080/02331934.2011.639374. Google Scholar

[6]

G. ButtazzoM. E. DrakhlinL. Freddi and E. Stepanov, Homogenization of optimal control problems for functional differential equations, J. Optim. Theory Appl., 93 (1997), 103-119. doi: 10.1023/A:1022649817825. Google Scholar

[7]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkh user, Boston, 2004. Google Scholar

[8]

A. Cernea, On the existence of viable solutions for a class of second order differential inclusions, Discuss. Math. Diff. Inc., Contr. Optim., 22 (2002), 67-78. doi: 10.7151/dmdico.1032. Google Scholar

[9]

F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics 264, Springer-Verlag London, 2013. doi: 10.1007/978-1-4471-4820-3. Google Scholar

[10]

V. F. Krotov, Methods of solution of variational problems on the basis of sufficient conditions of absolute minimum, Avtomat. i Telemekh., 23 (1962), 1571-1583. Google Scholar

[11]

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

[12]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. 1, Abstract Parabolic Systems: Continuous and Approximation Theories, Cambridge Univ. Press, 2000. Google Scholar

[13]

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

[14]

E. N. Mahmudov, Necessary and sufficient conditions for discrete and differential inclusions of elliptic type, J.Math. Anal. Appl., 323 (2006), 768-789. doi: 10.1016/j.jmaa.2005.10.069. Google Scholar

[15]

E. N. Mahmudov, Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions, Optim. Letters, 7 (2013), 871-891. doi: 10.1007/s11590-012-0460-1. Google Scholar

[16]

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

[17]

E. N. Mahmudov, Approximation and Optimization of Higher order discrete and differential inclusions, Nonlin. Diff. Equat. Appl. NoDEA, 21 (2014), 1-26. doi: 10.1007/s00030-013-0234-1. Google Scholar

[18]

E. N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal., 25 (2018), 293-318. Google Scholar

[19]

E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 495-525. Google Scholar

[20]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Indust. Manag. Optim., (2018), 13-35. doi: 10.3934/jimo.2018145. Google Scholar

[21]

E. N. Mahmudov, Optimization of fourth-order discrete-approximation inclusions, Appl. Math. Comput., 292 (2017), 19-32. doi: 10.1016/j.amc.2016.07.010. Google Scholar

[22]

E. N. Mahmudov, Optimization of boundary value problems for certain higher-order differential inclusions, J. Dyn. Control Syst., 25 (2019), 17-27. doi: 10.1007/s10883-017-9392-5. Google Scholar

[23]

E. N. Mahmudov, Optimal Control of Second Order Delay-Discrete and Delay Differential Inclusions with State Constraints, Evol. Equat. Cont. Theory (EECT), 7 (2018), 501-529. doi: 10.3934/eect.2018024. Google Scholar

[24]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I, Basic Theory; Ⅱ: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330 and 331, Springer, 2006. Google Scholar

[25]

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

[26]

L. S. Pontryagin, V. G. Boltyanskii, R. V Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, Inc., New York, London, Sydney; 1965. Google Scholar

[27]

A Dang QuangVu Thai Luan and Long Dang Quang, Iterative method for solving a fourth order differential equation with nonlinear boundary condition, Appl. Math. Sci., 4 (2010), 3467-3481. Google Scholar

[28]

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

show all references

References:
[1]

A. Auslender and J. Mechler, Second order viability problems for differential inclusions, J. Math. Anal. Appl., 181 (1994), 205-218. doi: 10.1006/jmaa.1994.1015. Google Scholar

[2]

D. Azzam-LaouirC. Castaing and L. Thibault, Three boundary value problems for second order differential inclusion in Banach spaces, Contr. Cybernet., 31 (2002), 659-693. Google Scholar

[3]

V. Barbu and T. Precupanu, Convex control problems in banach spaces, Convexity and Optimization in Banach Spaces, (2012), 233-364. doi: 10.1007/978-94-007-2247-7_4. Google Scholar

[4]

V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal controls, Proceed.Steklov Inst.Mathem., 169 (1986), 199-259. Google Scholar

[5]

D. Bors and M. Majewski, On Mayer problem for systems governed by second-order ODE, Optimization, 63 (2014), 239-254. doi: 10.1080/02331934.2011.639374. Google Scholar

[6]

G. ButtazzoM. E. DrakhlinL. Freddi and E. Stepanov, Homogenization of optimal control problems for functional differential equations, J. Optim. Theory Appl., 93 (1997), 103-119. doi: 10.1023/A:1022649817825. Google Scholar

[7]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkh user, Boston, 2004. Google Scholar

[8]

A. Cernea, On the existence of viable solutions for a class of second order differential inclusions, Discuss. Math. Diff. Inc., Contr. Optim., 22 (2002), 67-78. doi: 10.7151/dmdico.1032. Google Scholar

[9]

F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics 264, Springer-Verlag London, 2013. doi: 10.1007/978-1-4471-4820-3. Google Scholar

[10]

V. F. Krotov, Methods of solution of variational problems on the basis of sufficient conditions of absolute minimum, Avtomat. i Telemekh., 23 (1962), 1571-1583. Google Scholar

[11]

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

[12]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. 1, Abstract Parabolic Systems: Continuous and Approximation Theories, Cambridge Univ. Press, 2000. Google Scholar

[13]

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

[14]

E. N. Mahmudov, Necessary and sufficient conditions for discrete and differential inclusions of elliptic type, J.Math. Anal. Appl., 323 (2006), 768-789. doi: 10.1016/j.jmaa.2005.10.069. Google Scholar

[15]

E. N. Mahmudov, Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions, Optim. Letters, 7 (2013), 871-891. doi: 10.1007/s11590-012-0460-1. Google Scholar

[16]

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

[17]

E. N. Mahmudov, Approximation and Optimization of Higher order discrete and differential inclusions, Nonlin. Diff. Equat. Appl. NoDEA, 21 (2014), 1-26. doi: 10.1007/s00030-013-0234-1. Google Scholar

[18]

E. N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal., 25 (2018), 293-318. Google Scholar

[19]

E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 495-525. Google Scholar

[20]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Indust. Manag. Optim., (2018), 13-35. doi: 10.3934/jimo.2018145. Google Scholar

[21]

E. N. Mahmudov, Optimization of fourth-order discrete-approximation inclusions, Appl. Math. Comput., 292 (2017), 19-32. doi: 10.1016/j.amc.2016.07.010. Google Scholar

[22]

E. N. Mahmudov, Optimization of boundary value problems for certain higher-order differential inclusions, J. Dyn. Control Syst., 25 (2019), 17-27. doi: 10.1007/s10883-017-9392-5. Google Scholar

[23]

E. N. Mahmudov, Optimal Control of Second Order Delay-Discrete and Delay Differential Inclusions with State Constraints, Evol. Equat. Cont. Theory (EECT), 7 (2018), 501-529. doi: 10.3934/eect.2018024. Google Scholar

[24]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I, Basic Theory; Ⅱ: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330 and 331, Springer, 2006. Google Scholar

[25]

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

[26]

L. S. Pontryagin, V. G. Boltyanskii, R. V Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, Inc., New York, London, Sydney; 1965. Google Scholar

[27]

A Dang QuangVu Thai Luan and Long Dang Quang, Iterative method for solving a fourth order differential equation with nonlinear boundary condition, Appl. Math. Sci., 4 (2010), 3467-3481. Google Scholar

[28]

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

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