Pontryagin's Direct Method for Optimization Problems with Differential Inclusion

E. S. Polovinkin

Informatics and Automation, Optimal control and differential equations, Tome 304 (2019), pp. 257-272.

Résumé

We develop Pontryagin's direct variational method, which allows us to obtain necessary conditions in the Mayer extremal problem on a fixed interval under constraints on the trajectories given by a differential inclusion with generally unbounded right-hand side. The established necessary optimality conditions contain the Euler–Lagrange differential inclusion. The results are proved under maximally weak conditions, and very strong statements compared with the known ones are obtained; moreover, admissible velocity sets may be unbounded and nonconvex under a general hypothesis that the right-hand side of the differential inclusion is pseudo-Lipschitz. In the statements, we refine conditions on the Euler–Lagrange differential inclusion, in which neither the Clarke normal cone nor the limiting normal cone is used, as is common in the works of other authors. We also give an example demonstrating the efficiency of the results obtained.

Détail

Keywords: variational differential inclusion, adjoint Euler–Lagrange differential inclusion, necessary optimality conditions, derivatives of a multivalued mapping
Mots-clés : tangent cones, pseudo-Lipschitz condition.

Bibliographie

[1] Aubin J.-P., Frankowska H., Set-valued analysis, Birkhäuser, Boston, 1990 | MR | Zbl

[2] V. I. Blagodatskikh, “The maximum principle for differential inclusions”, Proc. Steklov Inst. Math., 166, 1986, 23–43 | Zbl | Zbl

[3] V. G. Boltyanskii, “The method of tents in the theory of extremal problems”, Russ. Math. Surv., 30:3 (1975), 1–54 | DOI | MR

[4] F. H. Clarke, Optimization and Nonsmooth Analysis, J. Wiley Sons, New York, 1983 | MR | Zbl

[5] Clarke F., Necessary conditions in dynamic optimization, Mem. AMS, 173, Amer. Math. Soc., Providence, RI, 2005 | MR

[6] Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth analysis and control theory, Grad. Texts Math., 178, Springer, New York, 1998 | MR | Zbl

[7] A. Ya. Dubovitskii and A. A. Milyutin, Necessary Conditions of Weak Extremum in the General Problem of Optimal Control, Nauka, Moscow, 1971 (in Russian)

[8] Ioffe A., “Euler–Lagrange and Hamiltonian formalisms in dynamic optimization”, Trans. Amer. Math. Soc., 349:7 (1997), 2871–2900 | DOI | MR | Zbl

[9] Kuratowski K., Topology, v. 1, Acad. Press, New York, 1966 | MR | Zbl

[10] Lee E.B., Markus L., Foundations of optimal control theory, J. Wiley Sons, New York, 1967 | MR | Zbl

[11] Michel P., Penot J.-P., “Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes”, C. r. Acad. sci. Paris. Sér. 1., 298 (1984), 269–272 | MR | Zbl

[12] B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988 (in Russian)

[13] Mordukhovich B.S., Variational analysis and generalized differentiation. I: Basic theory; II: Applications, Grundl. Math. Wiss., 330, 331, Springer, Berlin, 2006 | MR

[14] Polovinkin E.S., “The properties of continuity and differentiation of solution sets of Lipschitzean differential inclusions”, Modeling, estimation and control of systems with uncertainty, Prog. Syst. Control Theory, 10, eds. by G.B. Di Masi, A. Gombani, A.B. Kurzhansky, Birkhäuser, Boston, 1991, 349–360 | MR

[15] E. S. Polovinkin, “On the calculation of the polar cone of the solution set of a differential inclusion”, Proc. Steklov Inst. Math., 278, 2012, 169–178 | DOI | MR | Zbl

[16] E. S. Polovinkin, “Differential inclusions with measurable–pseudo-Lipschitz right-hand side”, Proc. Steklov Inst. Math., 283, 2013, 116–135 | DOI | MR | Zbl

[17] E. S. Polovinkin, Set-Valued Analysis and Differential Inclusions, Fizmatlit, Moscow, 2014 (in Russian)

[18] E. S. Polovinkin, “On the weak polar cone of the solution set of a differential inclusion with conic graph”, Proc. Steklov Inst. Math., 292, no. Suppl. 1, 2016, S253–S261 | DOI | MR | Zbl

[19] E. S. Polovinkin, “Differential inclusions with unbounded right-hand side and necessary optimality conditions”, Proc. Steklov Inst. Math., 291, 2015, 237–252 | DOI | MR | Zbl

[20] Polovinkin E.S., “Time optimum problems for unbounded differential inclusion”, IFAC-PapersOnLine, 48:25 (2015), 150–155 | DOI

[21] E. S. Polovinkin, “On the continuous dependence of trajectories of a differential inclusion on initial approximations”, Tr. Inst. Mat. Mekh. (Ekaterinburg), 25:1 (2019), 174–195 | MR

[22] E. S. Polovinkin and G. V. Smirnov, “An approach to the differentiation of many-valued mappings, and necessary conditions for optimization of solutions of differential inclusions”, Diff. Eqns., 22 (1986), 660–668 | MR | Zbl

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

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

[25] B. N. Pshenichnyi, Convex Analysis and Extremal Problems, Nauka, Moscow, 1980 (in Russian) | MR

[26] Sussmann H.J., “Geometry and optimal control”, Mathematical control theory, eds. by J. Baillieul, J.C. Willems, New York, Springer, 1998, 140–198 | MR | Zbl

[27] Vinter R., Optimal control, Birkhäuser, Boston, 2000 | MR | Zbl