# American Institute of Mathematical Sciences

September  2018, 7(3): 501-529. doi: 10.3934/eect.2018024

## Optimal control of second order delay-discrete and delay-differential inclusions with state constraints

 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  June 2017 Revised  February 2018 Published  July 2018

The present paper studies a new class of problems of optimal control theory with state constraints and second order delay-discrete (DSIs) and delay-differential inclusions (DFIs). The basic approach to solving this problem is based on the discretization method. Thus under the "regularity condition the necessary and sufficient conditions of optimality for problems with second order delay-discrete and delay-approximate DSIs are investigated. Then by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions the sufficient conditions of optimality for delay-DFIs, including the peculiar transversality ones, are proved. Here our main idea is the use of equivalence relations for subdifferentials of Hamiltonian functions and locally adjoint mappings (LAMs), which allow us to make a bridge between the basic optimality conditions of second order delay-DSIs and delay-discrete-approximate problems. In particular, applications of these results to the second order semilinear optimal control problem are illustrated as well as the optimality conditions for non-delayed problems are derived.

Citation: Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
##### References:
 [1] N. V. Antipina and V. A. Dykhta, Linear Lyapunov-Krotov functions and sufficient conditions for optimality in the form of the maximum principle, Russian Math. (Iz. VUZ), 46 (2002), 9-20.   Google Scholar [2] D. L. Azzam, A. Makhlouf and L. Thibault, Existence and relaxation theorem for a second order differential inclusion, Numer. Funct. Anal. Optim., 31 (2010), 1103-1119.  doi: 10.1080/01630563.2010.510982.  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, Sufficient conditions for optimality in problems with state constraints, Appl. Math. Optim., 7 (1981), 149-157.  doi: 10.1007/BF01442113.  Google Scholar [5] V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal control, Trudy Mat. Inst. Steklov., 169 (1985), 194-252, 255.   Google Scholar [6] A. 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Kipka and Y. S. Ledyaev, Optimal control of differential inclusions on manifolds, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 4455-4475.  doi: 10.3934/dcds.2015.35.4455.  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] 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 [14] 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 [15] E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar [16] E. N. Mahmudov, Duality in the problems of optimal-control for systems described by convex differential-inclusions with delay, Prob. Contr. Inform. Theory, 16 (1987), 411-422.   Google Scholar [17] E. N. Mahmudov, Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions, Nonlin. Anal., 67 (2007), 2966-2981.  doi: 10.1016/j.na.2006.09.054.  Google Scholar [18] 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 [19] E. N. Mahmudov, Optimal Control of Cauchy Problem for First-Order Discrete and Partial Differential Inclusions, J. Dyn. Contr. Syst., 15 (2009), 587-610.  doi: 10.1007/s10883-009-9073-0.  Google Scholar [20] 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 [21] E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 511-525.   Google Scholar [22] E. N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal., 25 (2018), 293-318.   Google Scholar [23] 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 [24] B. S. Mordukhovich and L. Wang, Optimal control of delay systems with differential and algebraic dynamic constraints, ESAIM: COCV, 11 (2005), 285-309.  doi: 10.1051/cocv:2005008.  Google Scholar [25] N. S. Papageorgiou and V. D. Rădulescu, 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] D. Q. A, V. T. Luan and D. Q. Long, Iterative method for solving a fourth order differential equation with nonlinear boundary condition, Appl. Math. Sci., 4 (2010), 3467-3481.   Google Scholar [27] S. H. Saker, R. P. Agarwal and D. O'Regan, Properties of solutions of fourth-order differential equations with boundary conditions, J. Inequalit. Appl., 278 (2013), 15pp. doi: 10.1186/1029-242X-2013-278.  Google Scholar [28] Q. Zhang and G. Li, Nonlinear boundary value problems for second order differential inclusions, Nonlin. Anal.: Theory, Methods Appl., 70 (2009), 3390-3406.  doi: 10.1016/j.na.2008.05.007.  Google Scholar [29] 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.  Google Scholar [30] D. G. Zill and M. R. Cullen, Differential Equations with Boundary-Value Problems, 5$^{nd}$ edition, Brooks/Cole, 2001. Google Scholar

show all references

##### References:
 [1] N. V. Antipina and V. A. Dykhta, Linear Lyapunov-Krotov functions and sufficient conditions for optimality in the form of the maximum principle, Russian Math. (Iz. VUZ), 46 (2002), 9-20.   Google Scholar [2] D. L. Azzam, A. Makhlouf and L. Thibault, Existence and relaxation theorem for a second order differential inclusion, Numer. Funct. Anal. Optim., 31 (2010), 1103-1119.  doi: 10.1080/01630563.2010.510982.  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, Sufficient conditions for optimality in problems with state constraints, Appl. Math. Optim., 7 (1981), 149-157.  doi: 10.1007/BF01442113.  Google Scholar [5] V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal control, Trudy Mat. Inst. Steklov., 169 (1985), 194-252, 255.   Google Scholar [6] A. Bressan, Differential inclusions and the control of forest fires, J. Diff. Equ. (special volume in honor of A. Cellina and J. Yorke), 243 (2007), 179-207.  doi: 10.1016/j.jde.2007.03.009.  Google Scholar [7] 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.  Google Scholar [8] P. Cannarsa and P. R. Wolenski, Semiconcavity of the value function for a class of differential inclusions, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 453-466.  doi: 10.3934/dcds.2011.29.453.  Google Scholar [9] S. Hu and N. S. Papageorgiou, Delay differential inclusions with constraints, Proceed.AMS, 123 (1995), 2141-2150.  doi: 10.1090/S0002-9939-1995-1257111-9.  Google Scholar [10] A. D. Ioffe and V. Tikhomirov, Theory of Extremal Problems, "Nauka", Moscow, 1974; English transl., North-Holland, Amsterdam, 1978.  Google Scholar [11] R. J. Kipka and Y. S. Ledyaev, Optimal control of differential inclusions on manifolds, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 4455-4475.  doi: 10.3934/dcds.2015.35.4455.  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] 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 [14] 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 [15] E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar [16] E. N. Mahmudov, Duality in the problems of optimal-control for systems described by convex differential-inclusions with delay, Prob. Contr. Inform. Theory, 16 (1987), 411-422.   Google Scholar [17] E. N. Mahmudov, Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions, Nonlin. Anal., 67 (2007), 2966-2981.  doi: 10.1016/j.na.2006.09.054.  Google Scholar [18] 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 [19] E. N. Mahmudov, Optimal Control of Cauchy Problem for First-Order Discrete and Partial Differential Inclusions, J. Dyn. Contr. Syst., 15 (2009), 587-610.  doi: 10.1007/s10883-009-9073-0.  Google Scholar [20] 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 [21] E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 511-525.   Google Scholar [22] E. N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints, J. Convex Anal., 25 (2018), 293-318.   Google Scholar [23] 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 [24] B. S. Mordukhovich and L. Wang, Optimal control of delay systems with differential and algebraic dynamic constraints, ESAIM: COCV, 11 (2005), 285-309.  doi: 10.1051/cocv:2005008.  Google Scholar [25] N. S. Papageorgiou and V. D. Rădulescu, 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] D. Q. A, V. T. Luan and D. Q. Long, Iterative method for solving a fourth order differential equation with nonlinear boundary condition, Appl. Math. Sci., 4 (2010), 3467-3481.   Google Scholar [27] S. H. Saker, R. P. Agarwal and D. O'Regan, Properties of solutions of fourth-order differential equations with boundary conditions, J. Inequalit. Appl., 278 (2013), 15pp. doi: 10.1186/1029-242X-2013-278.  Google Scholar [28] Q. Zhang and G. Li, Nonlinear boundary value problems for second order differential inclusions, Nonlin. Anal.: Theory, Methods Appl., 70 (2009), 3390-3406.  doi: 10.1016/j.na.2008.05.007.  Google Scholar [29] 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.  Google Scholar [30] D. G. Zill and M. R. Cullen, Differential Equations with Boundary-Value Problems, 5$^{nd}$ edition, Brooks/Cole, 2001. Google Scholar
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