# American Institute of Mathematical Sciences

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. [2] D. Azzam-Laouir, C. Castaing and L. Thibault, Three boundary value problems for second order differential inclusion in Banach spaces, Contr. Cybernet., 31 (2002), 659-693. [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. [4] V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal controls, Proceed.Steklov Inst.Mathem., 169 (1986), 199-259. [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. [6] G. Buttazzo, M. E. Drakhlin, L. 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. [7] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkh user, Boston, 2004. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] E. N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal., 25 (2018), 293-318. [19] E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 495-525. [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. [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. [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. [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. [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. [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. [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. [27] A Dang Quang, Vu 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. [28] Y. Zhou, V. 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.

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##### 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. [2] D. Azzam-Laouir, C. Castaing and L. Thibault, Three boundary value problems for second order differential inclusion in Banach spaces, Contr. Cybernet., 31 (2002), 659-693. [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. [4] V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal controls, Proceed.Steklov Inst.Mathem., 169 (1986), 199-259. [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. [6] G. Buttazzo, M. E. Drakhlin, L. 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. [7] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkh user, Boston, 2004. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] E. N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal., 25 (2018), 293-318. [19] E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 495-525. [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. [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. [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. [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. [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. [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. [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. [27] A Dang Quang, Vu 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. [28] Y. Zhou, V. 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.
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