Refined Euler--Lagrange Inclusion for an Optimal Control Problem with Discontinuous Integrand

S. M. Aseev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 34-63.

Résumé

We study a free-time optimal control problem for a differential inclusion with mixed-type functional in which the integral term contains the characteristic function of a given open set of “undesirable” states of the system. The statement of this problem can be viewed as a weakening of the statement of the classical optimal control problem with state constraints. Using the approximation method, we obtain first-order necessary optimality conditions in the form of the refined Euler–Lagrange inclusion. We also present sufficient conditions for their nondegeneracy and pointwise nontriviality and give an illustrative example.

Détail

Keywords: optimal control, differential inclusion, Pontryagin's maximum principle, refined Euler–Lagrange inclusion, state constraint, discontinuous integrand, risk zone.

Bibliographie

[1] A. V. Arutyunov, “Perturbations of extremal problems with constraints and necessary optimality conditions”, J. Sov. Math., 54:6 (1991), 1342–1400 | DOI

[2] A. V. Arutyunov, Optimality Conditions. Abnormal and Degenerate Problems, Kluwer, Dordrecht, 2000

[3] Arutyunov A.V., Aseev S.M., “Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints”, SIAM J. Control Optim., 35:3 (1997), 930–952 | DOI | MR

[4] Arutyunov A.V., Karamzin D.Yu., Pereira F.L., “The maximum principle for optimal control problems with state constraints by R. V. Gamkrelidze: Revisited”, J. Optim. Theory Appl., 149:3 (2011), 474–493 | DOI | MR

[5] Aseev S.M., “Methods of regularization in nonsmooth problems of dynamic optimization”, J. Math. Sci., 94:3 (1999), 1366–1393 | DOI | MR

[6] S. M. Aseev, “Extremal problems for differential inclusions with state constraints”, Proc. Steklov Inst. Math., 233 (2001), 1–63

[7] S. M. Aseev, “Optimization of dynamics of a control system in the presence of risk factors”, Tr. Inst. Mat. Mekh. (Ekaterinburg), 23:1 (2017), 27–42 | MR

[8] S. M. Aseev, “On an optimal control problem with discontinuous integrand”, Proc. Steklov Inst. Math., 304:Suppl. 1 (2019), S3–S13 | DOI | MR

[9] Aseev S.M., “An optimal control problem with a risk zone”, Large-scale scientific computing: 11th Int. Conf., LSSC 2017 (Sozopol, 2017), Lect. Notes Comput. Sci., 10665, Springer, Cham, 2018, 185–192 | DOI

[10] Aseev S.M., “A problem of dynamic optimization in the presence of dangerous factors”, Stability, control and differential games: Proc. Int. Conf. SCDG2019 (Yekaterinburg, 2019), Lect. Notes Control Inf. Sci. – Proc., Springer, Cham, 2020, 273–281

[11] S. M. Aseev and A. I. Smirnov, “The Pontryagin maximum principle for the problem of optimally crossing a given domain”, Dokl. Math., 69:2 (2004), 243–245 | MR

[12] S. M. Aseev and A. I. Smirnov, “Necessary first-order conditions for optimal crossing of a given region”, Comput. Math. Model., 18:4 (2007), 397–419 | DOI | MR

[13] Cesari L., Optimization—theory and applications: Problems with ordinary differential equations, Springer, New York, 1983

[14] Clarke F.H., “The Euler–Lagrange differential inclusion”, J. Diff. Eqns., 19 (1975), 80–90 | DOI

[15] F. H. Clarke, Optimization and Nonsmooth Analysis, J. Wiley Sons, New York, 1983

[16] Clarke F., Functional analysis, calculus of variations and optimal control, Grad. Texts Math., 264, Springer, London, 2013 | DOI

[17] Ferreira M.M.A., Vinter R.B., “When is the maximum principle for state constrained problems nondegenerate?”, J. Math. Anal. Appl., 187:2 (1994), 438–467 | DOI | MR

[18] Fontes F.A.C.C., Frankowska H., “Normality and nondegeneracy for optimal control problems with state constraints”, J. Optim. Theory Appl., 166:1 (2015), 115–136 | DOI | MR

[19] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 2009

[20] B. Sh. Mordukhovich, “Maximum principle in the problem of time optimal response with nonsmooth constraints”, J. Appl. Math. Mech., 40 (1976), 960–969 | DOI | MR

[21] Mordukhovich B.Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988

[22] B. Sh. Mordukhovich, “Optimal control of difference, differential, and differential–difference inclusions”, J. Math. Sci., 100:6 (2000), 2613–2632 | DOI | MR

[23] I. P. Natanson, Theory of Functions of a Real Variable, Dover Publ., Mineola, NY, 2016

[24] E. S. Polovinkin and G. V. Smirnov, “Time-optimal problem for differential inclusions”, Diff. Eqns., 22:8 (1986), 940–952 | MR

[25] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Pergamon, Oxford, 1964

[26] Pshenichnyi B.N., Ochilov S., “O zadache optimalnogo prokhozhdeniya cherez zadannuyu oblast”, Kibernetika i vychisl. tekhnika, 99 (1993), 3–8

[27] Pshenichnyi B.N., Ochilov S., “Ob odnoi spetsialnoi zadache optimalnogo bystrodeistviya”, Kibernetika i vychisl. tekhnika, 101 (1994), 11–15

[28] Schneider R., “Equivariant endomorphisms of the space of convex bodies”, Trans. Amer. Math. Soc, 194 (1974), 53–78 | DOI | MR

[29] A. I. Smirnov, “Necessary optimality conditions for a class of optimal control problems with discontinuous integrand”, Proc. Steklov Inst. Math., 262 (2008), 213–230 | DOI | MR

[30] Smirnov G.V., Introduction to the theory of differential inclusions, Grad. Stud. Math., 41, Amer. Math. Soc., Providence, RI, 2002

[31] Vinter R., Optimal control, Birkhäuser, Boston, 2000