M. I. Sumin. On the regularization of the classical optimality conditions in convex optimal control problems ... P. 252-269

We consider a regularization of the classical optimality conditions (COCs) in a convex optimal control problem for a linear system of ordinary differential equations with a pointwise state equality constraint and a finite number of functional constraints in the form of equalities and inequalities. The set of admissible controls of the problem is traditionally embedded in the space of square integrable functions. However, the objective functional is not, generally speaking, strongly convex. The proof of regularized COCs is based on the use of two regularization parameters. One of them is “responsible” for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the objective functional of the original problem. The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized COCs: (1) are formulated as theorems on the existence of minimizing approximate solutions in the original problem with the simultaneous constructive presentation of their specific representatives; (2) are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; (3) are sequential generalizations of their classical counterparts and retain their general structure; (4) “overcome” the properties of ill-posedness of COCs and are regularizing algorithms for optimization problems.

Keywords: convex optimal control, convex programming, minimizing sequence, regularizing algorithm, Lagrange principle, Pontryagin maximum principle, dual regularization.

REFERENCES

1.   Warga J. Optimal control of differential and functional equations. N Y: Acad. Press, 1972, 531 p. ISBN: 0127351507 . Translated to Russian under the title Optimal’noe upravlenie differentsial’nymi i funktsional’nymi uravneniyami. Moscow: Nauka Publ., 1977, 624 p.

2.   Golshtein E.G. Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya [Duality theory in mathematical programming and its applications]. Moscow: Nauka Publ., 1971, 352 p.

3.   Sumin M.I. Parametric dual regularization for an optimal control problem with pointwise state constraints. Comput. Math. Math. Phys., 2009, vol. 49, no. 12, pp. 1987–2005.
doi: 10.1134/S096554250912001X .

4.   Sumin M.I. Regularized parametric Kuhn–Tucker theorem in a Hilbert space. Comput. Math. Math. Phys., 2011, vol. 51, no. 9, pp. 1489–1509. doi: 10.1134/S0965542511090156 .

5.   Sumin M.I. Duality-based regularization in a linear convex mathematical programming problem. Comput. Math. Math. Phys., 2007, vol. 47, no. 4, pp. 579–600. doi: 10.1134/S0965542507040045 .

6.   Sumin M.I. Stable sequential convex programming in a Hilbert space and its application for solving unstable problems. Comput. Math. Math. Phys., 2014, vol. 54, no. 1, pp. 22–44.
doi: 10.1134/S0965542514010138 .

7.   Arutyunov A.V. Optimality conditions. Abnormal and degenerate problems. Ser. Mathematics and Its Applications (Dordrecht), vol. 526, Dordrecht: Kluwer Academic Publ., 2000, 300 p. doi: 10.1007/978-94-015-9438-7 . Original Russian text published in Arutyunov A.V. Usloviya ekstremuma. Anormal’nye i vyrozhdennye zadachi. Moscow: Faktorial Publ., 1997, 255 p. ISBN: 5-88688-015-1 .

8.   Milyutin A.A., Dmitruk A.V., Osmolovskii N.P. Printsip maksimuma v optimal’nom upravlenii [Maximum principle in optimal control]. Moscow: Center of Applied Investigations at the Faculty of Mechanics and Mathematics in MSU, 2004, 168 p.

9.   Sumin M.I. On the stable sequential Lagrange principle in convex programming and its application for solving unstable problems. Trudy Inst. Mat. Mekh. UrO RAN, 2013, vol. 19, no. 4, pp. 231–240 (in Russian).

10.   Kuterin F.A., Sumin M.I. The regularized iterative Pontryagin maximum principle in optimal control. I. Optimization of a lumped system. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp‘yuternye Nauki, 2016, vol. 26, no. 4, pp. 474–489 (in Russian). doi: 10.20537/vm160403 .

11.   Breitenbach T., Borzi A. A sequential quadratic hamiltonian method for solving parabolic optimal control problems with discontinuous cost functionals. J. Dyn. Control Syst., 2019, vol. 25, no. 3, pp. 403–435. doi: 10.1007/s10883-018-9419-6 .

12.   Breitenbach T., Borzi A. On the SQH scheme to solve nonsmooth PDE optimal control problems. Numerical Functional Analysis and Optimization, 2019, vol. 40, no. 13, pp. 1489–1531.
doi: 10.1080/01630563.2019.1599911 .

13.   Vasil’ev F.P. Metody optimizatsii [Optimization methods]. Moscow: MTsNMO Publ., 2011. Vol. 1: 620 p., ISBN: 978-5-94057-707-2 ; Vol. 2: 433 p., ISBN: 978-5-94057-708-9 .

14.   Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal control. N Y: Plenum Press, 1987, 309 p. doi: 10.1007/978-1-4615-7551-1 . Original Russian text published in Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal’noe upravlenie. Moscow: Nauka Publ., 1979, 432 p.

Received March 24, 2020

Revised May 2, 2020

Accepted May 18, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research (projects no. 19-07-00782_a, 20-01-00199_a, 20-52-00030 Bel_a).

Mikhail Iosifovich Sumin, Dr. Phys.-Math. Sci., Prof., Tambov State University, Tambov, 392000 Russia; Nizhnii Novgorod State University, Nizhnii Novgorod, 603950 Russia,
e-mail: m.sumin@mail.ru.

Cite this article as: M.I.Sumin. On the regularization of the classical optimality conditions in convex optimal control problems. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 252–269.