Positional strengthenings of the maximum principle and sufficient optimality conditions

V. A. Dykhta

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 2, pp. 73-86

Résumé

We derive nonlocal necessary optimality conditions, which efficiently strengthen the classical Pontryagin maximum principle and its modification obtained by B.Kaskosz and S.Lojasiewicz as well as our previous result of a similar kind named the “feedback minimum principle”. The strengthening of the feedback minimum principle (and, hence, of the Pontryagin principle) is owing to the employment of two types of feedback controls “compatible” with a reference trajectory (i.e., producing this trajectory as a Caratheodory solution). In each of the versions, the strengthened feedback minimum principle states that the optimality of a reference process implies the optimality of its trajectory in a certain family of variational problems generated by adjoint trajectories of the original and compatible controls. The basic construction of the feedback minimum principle - a perturbation of a solution to the adjoint system - is employed to prove an exact formula for the increment of the cost functional. We use this formula to obtain sufficient conditions for the strong and global minimum of Pontryagin's extremals. These conditions are much milder than their known analogs, which require the convexity in the state variable of the functional and of the lower Hamiltonian. Our study is focused on a nonlinear smooth Mayer problem with free terminal states. All assertions are illustrated by examples.

Détail

Keywords: maximum principle, extremal, adjoint trajectory, necessary and sufficient conditions, feedback controls.

Bibliographie

[1] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mischenko, Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 388 pp. | MR

[2] Dykhta V.A., “Variatsionnye usloviya optimalnosti s pozitsionnymi upravleniyami spuska, usilivayuschie printsip maksimuma”, Izv. Irkut. gos. un-ta. Ser. Matematika, 8 (2014), 86–103

[3] Dykhta V.A., “Slabo monotonnye resheniya neravenstva Gamiltona - Yakobi i usloviya optimalnosti s pozitsionnymi upravleniyami”, Avtomatika i telemekhanika, 2014, no. 5, 31–49 | MR | Zbl

[4] Dykhta V.A., “Nestandartnaya dvoistvennost i nelokalnye neobkhodimye usloviya optimalnosti v nevypuklykh zadachakh optimalnogo upravleniya”, Avtomatika i telemekhanika, 2014, no. 11, 19–37

[5] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985, 518 pp. | MR

[6] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t kompyuternykh issledovanii, M.; Izhevsk, 2003, 336 pp.

[7] Klark F., Ledyaev Yu.S., Subbotin A.I., “Universalnoe pozitsionnoe upravlenie i proksimalnoe pritselivanie v zadachakh upravleniya v usloviyakh vozmuscheniya i differentsialnykh igrakh”, Tr. MIAN, 224 (1999), 165–186 | MR

[8] F.H. Clarke, Yu. SLedyaev, R.J. Stern, P.R. Wolenski, Nonsmooth analysis and control theory, Springer-Verlag, N.Y., 1998, 276 pp. | MR | Zbl

[9] F.H. Clarke, Yu. SLedyaev, R.J. Stern, P.R. Wolenski, “Qualitative properties of trajectories of control systems: a survey”, J. Dynam. Control Systems, 1:1 (1995), 1–48 | DOI | MR | Zbl

[10] Warga J., “A second order condition that strengthens Pontryagin's maximum principle”, J. Differential Equations, 28:2 (1978), 284–307 | DOI | MR | Zbl

[11] Kaskosz B., Lojasiewicz S., “A maximum principle for generalized control”, Nonlinear Analysis: Theory, Methods and Appl., 9:2 (1985), 109–130 | DOI | MR | Zbl

[12] Kaskosz B., “Extremality, controllability, and abundant subsets of generalized control systems”, J. Optimization Theory Appl., 101:1 (1999), 73–108 | DOI | MR | Zbl

[13] Frankowska H., Kaskosz B., “Linearization and boundary trajectories of nonsmooth control systems”, Can. J. Math., XI:3 (1988), 589–609 | DOI | MR

[14] Sussmann H.J., “A strong version of the Lojasiewicz maximum principle”, Optimal Control of Differential Equations, Lecture Notes in Pure and Applied Mathematics, ed. N.H. Pavel, M. Dekker Inc., N.Y., 1994, 1–17 | MR

[15] Loewen P.D., Vinter R.B., “Pontryagin-type necessary conditions for differential inclusion problems”, Systems Control Lett, 9:9 (1997), 263–265 | MR

[16] Artstein Z., “Pontryagin maximum principle revisited with feedbacks”, Eur. J. Control, 17:1 (2011), 46–54 | DOI | MR | Zbl

[17] Clarke F., Functional analysis, calculus of variations and optimal control, Graduate Texts in Mathematics, 264, Springer-Verlag, London, 2013, 591 pp. | DOI | MR | Zbl

[18] Nikolskii M.S., “O dostatochnosti printsipa maksimuma Pontryagina v nekotorykh optimizatsionnykh zadachakh”, Vestn. Mosk. un-ta. Ser. 15: Vychisl. matematika i kibernetika, 2005, no. 1, 35–43 | MR | Zbl

[19] Antipina N.V., Dykhta V.A., “Lineinye funktsii Lyapunova - Krotova i dostatochnye optimalnosti v forme printsipa maksimuma”, Izv. vuzov. Matematika, 2002, no. 12, 11–22 | MR | Zbl

[20] Krotov V.F., Global methods in optimal control theory, Monographs and Textbooks in Pure and Applied Mathematics, 195, Marcel Dekker, N.Y., 1996, 384 pp. | MR | Zbl

[21] Dykhta V.A., “Analiz dostatochnykh uslovii optimalnosti s mnozhestvom funktsii tipa Lyapunova”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16:5 (2010), 66–75

[22] Gabasov R., Kirillova F.M., Printsip maksimuma v teorii optimalnogo upravleniya, Editorial URSS, M., 2011, 272 pp. | MR