Optimal control of higher order differential inclusions with functional constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 37.

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (P$$); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (P$$). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (P$$) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Reçu le :
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/cocv/2019018
Classification : 49k20, 49k24, 49J52, 49M25, 90C31
Mots-clés : Higher order differential inclusions, complementary slackness, Euler–Lagrange, approximation, transversality, set-valued 
@article{COCV_2020__26_1_A37_0,
     author = {Mahmudov, Elimhan N.},
     title = {Optimal control of higher order differential inclusions with functional constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2019018},
     mrnumber = {4116680},
     zbl = {1447.49039},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2019018/}
}
TY  - JOUR
AU  - Mahmudov, Elimhan N.
TI  - Optimal control of higher order differential inclusions with functional constraints
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2020
VL  - 26
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2019018/
DO  - 10.1051/cocv/2019018
LA  - en
ID  - COCV_2020__26_1_A37_0
ER  - 
%0 Journal Article
%A Mahmudov, Elimhan N.
%T Optimal control of higher order differential inclusions with functional constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2020
%V 26
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2019018/
%R 10.1051/cocv/2019018
%G en
%F COCV_2020__26_1_A37_0
Mahmudov, Elimhan N. Optimal control of higher order differential inclusions with functional constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 37. doi : 10.1051/cocv/2019018. http://www.numdam.org/articles/10.1051/cocv/2019018/

[1] D. Affane and D.L. Azzam, Almost convex valued perturbation to time optimal control sweeping processes. ESAIM: COCV 23 (2017) 1–12. | Numdam | MR | Zbl

[2] A. Auslender and J. Mechler, Second order viability problems for differential inclusions. J. Math. Anal. Appl. 181 (1994) 205–218. | DOI | MR | Zbl

[3] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monographs in Mathematics. Springer, Berlin (2003). | MR | Zbl

[4] 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 | MR | Zbl

[5] M.H.A. Biswas and M.R. Pinho, A maximum principle for optimal control problems with state and mixed constraints. ESAIM: COCV 21 (2015) 939–957. | Numdam | MR | Zbl

[6] L. Bourdin and E. Trélat, Pontryagin Maximum Principle for finite dimensional nonlinear optimal control problems on time scales. SIAM J. Control Optim. 51 (2013) 3781–3813. | DOI | MR | Zbl

[7] A. Bressan and Tao Wang, Equivalent formulation and numerical analysis of a fire confinement problem. ESAIM: COCV 16 (2010) 974–1001. | Numdam | MR | Zbl

[8] P. Cannarsa, H. Frankowska and T. Scarinci, Sensitivity relations for the Mayer problem with differential inclusions. ESAIM: COCV 21 (2015) 789–814. | Numdam | MR | Zbl

[9] F.H. Clarke, Optimization and Nonsmooth Analysis. John Wiley and Sons Inc., New York (1983). | MR | Zbl

[10] T. Donchev and M. Quincampoix, Nonemptiness of viability kernels for infinite-dimensional differential inclusions. Appl. Math. Lett. 16 (2003) 1195–1199. | DOI | MR | Zbl

[11] R. Hilscher and V. Zeidan, Discrete optimal control: second order optimality conditions. J. Differ. Equ. Appl. 8 (2002) 875–896. | DOI | MR | Zbl

[12] A.D. Ioffe and V. Tikhomirov, Theory of Extremal Problems. Nauka, Moscow (1974); English translation, North-Holland, Amsterdam (1978). | MR

[13] P.D. Loewen and R.T. Rockafellar, Optimal control of unbounded differential inclusions. SIAM J. Control Optim. 32 (1994) 442–470. | DOI | MR | Zbl

[14] V. Lupulescu, Viable solutions for second order nonconvex functional differential inclusions. Electron. J. Differ. Equ. 2005 (2005) 1–11. | MR | Zbl

[15] 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 | MR | Zbl

[16] E.N. Mahmudov, Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions. J. Nonlinear Anal. 67 (2007) 2966–2981. | DOI | MR | Zbl

[17] E.N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions. Elsevier, Boston, USA (2011). | MR | Zbl

[18] E.N. Mahmudov, Approximation and Optimization of Higher order discrete and differential inclusions. Nonlinear Differ. Equ. Appl. (NoDEA) 21 (2014) 1–26. | DOI | MR | Zbl

[19] E.N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions. Pac. J. Optim. 11 (2015) 495–525. | MR | Zbl

[20] E.N. Mahmudov, Free time optimization of higher order differential inclusions with endpoint constraints. Appl. Anal. 97 (2017) 2071–2084. | DOI | MR | Zbl

[21] E.N. Mahmudov, Optimization of second order differential inclusions with Boundary value conditions. J. Nonlinear Convex Anal. (JNCA) 18 (2017) 1653–1664. | MR | Zbl

[22] E.N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints. J. Convex Anal. 25 (2018) 1–26. | MR | Zbl

[23] E.N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville type differential inclusions. J. Optim. Theory Appl. 177 (2018) 345–375. | DOI | MR | Zbl

[24] E.N. Mahmudov, Free time optimization of second-order differential inclusions with endpoint constraints. J. Dyn. Control Syst. 24 (2018) 129–143. | DOI | MR | Zbl

[25] L. Marco and J.A. Murillo, Lyapunov functions for second order differential inclusions: a viability approach. J. Math. Anal. Appl. 262 (2001) 339–354. | DOI | MR | Zbl

[26] B.S. Mordukhovich and L. Wang, Optimal control of delay systems with differential and algebraic dynamic constraints. ESAIM: COCV 11 (2005) 285–309. | Numdam | MR | Zbl

[27] B.S. Mordukhovich, Optimal control of semilinear unbounded evolution inclusions with functional constraints. J. Optim Theory Appl. 167 (2015) 821–841. | DOI | MR | Zbl

[28] M.R. Pinho, M.M. Ferreira and F. Fontes, Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems. ESAIM: COCV 11 (2005) 614–632. | Numdam | MR | Zbl

Cité par Sources :