M.I. Sumin. Regularization of the Pontryagin maximum principle in a convex optimal boundary control problem for a parabolic equation with an operator equality constraint ... P. 221-237

We consider the regularization of the classical optimality conditions — the Lagrange principle (LP) and the Pontryagin maximum principle (PMP) — in a convex optimal control problem for a parabolic equation with an operator equality constraint and with a boundary control. The set of admissible controls of the problem is traditionally embedded into the space of square-summable functions. However, the objective functional is not, generally speaking, strongly convex. The derivation of regularized LP and PMP 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 LP and PMP is the stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized LP and PMP are formulated as existence theorems in the original problem of minimizing approximate solutions consisting of minimals of its regular Lagrange function. They “overcome” the ill-posedness properties of the LP and PMP and are regularizing algorithms for solving the optimal control problem. Particular attention is paid to the proof of the PMP in the problem of minimizing the regular Lagrange function and obtaining on this basis the regularized PMP in the original optimal control problem as a consequence of the regularized LP.

Keywords: convex optimal control, parabolic equation, operator constraint, boundary control, minimizing sequence, regularizing algorithm, Lagrange principle, Pontryagin maximum principle, dual regularization

Received January 29, 2021

Revised February 13, 2021

Accepted March 1, 2021

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

REFERENCES

1.   Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishechenko E.F. The mathematical theory of optimal processes. N Y; London: John Wiley & Sons, 1962, 360 p. doi: 10.1002/zamm.19630431023 . Original Russian text published in Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Fizmatgiz Publ., 1961, 391 p.

2.   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.

3.   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 .

4.   Sumin M.I. Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems. Trudy Inst. Mat. Mekh. UrO RAN, 2019, vol. 25, no. 1, pp. 279–296 (in Russian). doi: 10.21538/0134-4889-2019-25-1-279-296 

5.   Sumin M.I. On regularization of the classical optimality conditions in convex optimal control problems. Trudy Inst. Mat. Mekh. UrO RAN, 2020, vol. 26, no. 2, pp. 252–269 (in Russian). doi: 10.21538/0134-4889-2020-26-2-252-269 

6.   Tikhonov A.N., Arsenin V.Ya. Methods for solutions of ill-posed problems. N Y: Wiley, 1977, 258 p. ISBN: 0470991240 . Original Russian text published in Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach. Moscow: Nauka Publ., 1986, 288 p.

7.   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.

8.   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.

9.   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 

10.   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 

11.   Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N. Linear and quasilinear equations of parabolic type. Providence, R.I.: AMS, 1968, 648 p. ISBN: 978-0-8218-1573-1 . Original Russian text published in Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N. Lineinye i kvazilineinye uravneniya parabolicheskogo tipa. Moscow: Nauka Publ., 1967, 736 p.

12.   Plotnikov V.I. Uniqueness and existence theorems and apriori properties of generalized solutions. Sov. Math., Dokl., 1965, vol. 6, pp. 1405–1407.

13.   Raymond J.P., Zidani H. Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim., 1999, vol. 39, no. 2, pp. 143–177. doi: 10.1007/s002459900102 

14.    Casas E. Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim., 1997, vol. 35, pp. 1297–1327. doi: 10.1137/S0363012995283637 

15.   Sumin M.I. A regularized gradient dual method for the inverse problem of a final observation for a parabolic equation. Comput. Math. Math. Phys., 2004, vol. 44, no. 11, pp. 1903–1921.

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, 2021, vol. 27, no. 2, pp. 221–237.