Optimization of neutral functional-differential inclusions. (English) Zbl 1317.49033

Summary: A Bolza problem of optimal control theory with a varying time interval given by convex, nonconvex functional-differential inclusions \((P_N)\), \((P_V)\) is considered. Our main goal is to derive sufficient optimality conditions for neutral functional-differential inclusions, which contain time delays in both state and velocity variables. Both state and endpoint constraints are involved. The presence of constraint conditions implies jump conditions for the conjugate variable, which are typical for such problems. Sufficient conditions under the \(t_1\)-transversality condition are proved incorporating the Euler-Lagrange- and Hamiltonian-type inclusions. Supplementary problems with discrete and discrete approximation inclusions \((P_D)\), \((P_{DA})\) are considered and necessary, and sufficient conditions are given. The basic concept for obtaining optimality conditions are locally adjoint mappings and especially proved equivalence theorems. Furthermore, the application of these results is demonstrated by solving some illustrative examples.


49K21 Optimality conditions for problems involving relations other than differential equations
34K09 Functional-differential inclusions
34A60 Ordinary differential inclusions
54C60 Set-valued maps in general topology
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[1] Agrachev AA, Sachkov YL. Control theory from the geometric viewpoint, control theory and optimization II. Berlin: Springer; 2004.
[2] Aubin JR, Cellina H. Differential inclusion, Springer,Grudlehnender Mat., Wiss.,1984. · Zbl 0734.49009
[3] Viorel Barbu. Convexity and optimization in Banach Spasces, Sijthoff (1978). 2nd ed. Dordrecht: D.Reiderel; 1986.
[4] Barbu V. Analysis and control of infinite dimensional systems. New York: Academic; 1983.
[5] Clarke FH. Optimization and nonsmooth analysis. New York: Wiley-Interscience; 1983. · Zbl 0582.49001
[6] Clarke, FH; Wolenski, PR, Necessary conditions for functional differential inclusions, Appl Math Optim, 34, 34-51, (1996) · Zbl 0877.49022
[7] Clarke, FH, Hamiltonian analysis of the generalized problem of Bolza, Trans Am Math Soc, 301, 385-400, (1987) · Zbl 0621.49011
[8] Demyanov VF, Vasilev LV. Nondifferentiable optimization. New York: Optimization Software; 1985. · Zbl 0572.00008
[9] Gabasov R, Kirillova FM. Maximum principle in theory of optimal control. Minsk: Nauka i Teknika; 1974.
[10] Hale J. Theory of functional differential equations. New York: Springer; 1977. · Zbl 0352.34001
[11] Haddad G. Topological properties of the sets of solutions for functional differential inclusions, Nonlinear Analysis, 2001;5(12):1349–1366. · Zbl 0496.34041
[12] Haddad, G; Lasry, JM, Periodic solutions of functional differential inclusions and fixed points of σ-selectionable correspondences, J Math Anal Appl, 96, 295-312, (1983) · Zbl 0539.34031
[13] Hernandez, E; Henriquez, HR, Existence results for partial neutral functional differential equations with unbounded delay, J Math Anal Appl, 221, 452-75, (1998) · Zbl 0915.35110
[14] Ioffe AD, Tikhomirov VM. Theory of extremal problems, Nauka, Moscow, 1974 (in Russian); English transl., North-Holland, Amsterdam; 1979. · Zbl 1208.49031
[15] Loewen, PD; Rockafellar, RT, The adjoint arc in nonsmooth optimization, Trans Am Math Soc, 325, 39-72, (1991) · Zbl 0734.49009
[16] Kolmanovskii VP, Shaikhet LE. Control of systems with aftereffect. New York: Academic; 1996.
[17] Mahmudov EN. Approximation and optimization of discrete and differential inclusions. Waltham, MA: Elsevier; 2011
[18] Mahmudov, EN, Optimal control of Cauchy problem for first-order discrete and partial differential inclusions, J Dyn Control Syst, 15, 587-610, (2009) · Zbl 1203.49029
[19] Mahmudov EN. Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions. Nonlinear Anal, 2007;67(10):2966–2981. · Zbl 1117.49023
[20] Mahmudov, EN, Sufficient conditions for optimality for differential inclusions of parabolic type and duality, J Glob Optim, 41, 31-42, (2008) · Zbl 1149.49027
[21] Mahmudov, EN, On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type, J Math Anal Appl, 307, 628-40, (2005) · Zbl 1069.49026
[22] Mahmudov, EN, The optimality principle for discrete and first order partial differential inclusions, J Math Anal Appl, 308, 605-19, (2005) · Zbl 1074.49006
[23] Mahmudov, EN, Optimization of differential inclusions of Bolza type with state constraints and duality, Math J, 7, 21-38, (2005) · Zbl 1125.49019
[24] Mordukhovich BS. Variational analysis and generalised differentiation I, II, Grundlehren der mathematischen Wissenschaften. 2006 Vol. 330,331. · Zbl 1125.49019
[25] Mordukhovich, BS; Wang, L, Optimal control of neutral functional-differential inclusions linear in velocities, TEMA Tend Mat Apl Comput, 5, 1-15, (2004) · Zbl 1208.49031
[26] Medhin, NG, On optimal control of functional-differential systems, J Optim Theory Appl, 85, 363-76, (1995) · Zbl 0826.49016
[27] Outrata, J; Mordukhovich, BS, Coderivative analysis of quasivariational inequalities with applications to stability and optimization, SIAM J Optim, 18, 389-412, (2007) · Zbl 1145.49012
[28] Pontryagin LS, Boltyanskii VG, Gamkrelidze RV,Mishchenko EF. The mathematical theory of optimal processes. New York: Wiley; 1965.
[29] Pshenichnyi BN. Convex analysis and extremal problems. Moscow: Nauka; 1980 (Russian). · Zbl 0477.90034
[30] Rowland, JDL; Vinter, RB, Dynamic optimization problems with free time and active state constraints, SIAM J Control Optim, 31, 677-91, (1993) · Zbl 0779.49028
[31] Rubinov AM. Superlinear multivalued mappings and their applications to economical-mathematical problems. Leningrad: Nauka; 1980 (in Russian).
[32] Richard, L; Yung, CH, Optimality conditions and duality for a non-linear time delay control problem, Optim Control Appl Methods, 18, 327-40, (1997) · Zbl 0884.49013
[33] Hong, S, Boundary-value problems for first and second order functional differential inclusions, Electron J Differ Equ, 32, 1-10, (2003)
[34] Vinter, R; Zheng, HH, Necessary conditions for free end-time measurably time dependent optimal control problems with state constraints, Set-Valued Anal, 8, 11-29, (2000) · Zbl 0967.49017
[35] Zhu, QJ, Necessary optimality conditions for nonconvex differential inclusion with endpoint constraints, J Diff Equat, 124, 186-204, (1996) · Zbl 0899.49010
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