On regularization of the classical optimality conditions in the convex optimization problems for Volterra-type systems with operator constraints

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Abstract

We consider the regularization of classical optimality conditions (COCs) — the Lagrange principle (LP) and the Pontryagin maximum principle (PMP) — in a convex optimal control problem with an operator equality-constraint and functional inequality-constraints. The controlled system is specified by a linear functional-operator equation of the second kind of general form in the space L2m, the main operator on the right side of the equation is assumed to be quasinilpotent.The objective functional of the problem is only convex (perhaps not strongly convex). Obtaining regularized COCs is based on the dual regularization method. In this case, two regularization parameters are used, one of which is “responsible” for the regularization of the dual problem, the other is contained in the strongly convex regularizing Tikhonov addition to the target functional of the original problem, thereby ensuring the correctness of the problem of minimizing the Lagrange function. The main purpose of regularized LP and PMP is the stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs: 1) are formulated as existence theorems for minimizing approximate solutions in the original problem with a simultaneous constructive representation of these solutions; 2) expressed in terms of regular classical functions of Lagrange and Hamilton–Pontryagin; 3) “overcome” the properties of the ill-posedness of the COCs and provide regularizing algorithms for solving optimization problems. Based on the perturbation method, an important property of the regularized COCs obtained in the work is discussed in sufficient detail, namely that “in the limit” they lead to their classical analogues. As an application of the general results obtained in the paper, a specific example of an optimal control problem associated with an integro-differential equation of the transport equation type is considered, a special case of which is a certain final observation problem. 

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V. I. Sumin

Derzhavin Tambov State University

Author for correspondence.
Email: v_sumin@mail.ru
Russian Federation, Tambov

M. I. Sumin

Lobachevskii Nizhnii Novgorod State University

Email: m.sumin@mail.ru
Russian Federation, Nizhnii Novgorod

References

  1. Alekseev, V.M. Optimal Control / V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin. — New York : Plenum Press, 1987.
  2. Avakov, E.R. Lagrange’s principle in extremum problems with constraints / E.R. Avakov, G.G. Magaril-Il’yaev, V.M. Tikhomirov // Russ. Math. Surveys. — 2013. — V. 68, № 3. — P. 401–433.
  3. Arutyunov, A.V. Printsip maksimuma Pontryagina. Dokazatel‘stvo i prilozheniya / A.V. Arutyunov, G.G. Magaril-Il’yaev, V.M. Tikhomirov. — Moscow : Faktorial Press, 2006. — 144 p. [in Russian]
  4. Gamkrelidze, R.V. History of the discovery of the Pontryagin maximum principle / R.V. Gamkrelidze // Proc. Steklov Inst. Math. — 2019. — V. 304. — P. 1–7.
  5. Il-posed problems in the natural science : coll. art. / By eds. A.N. Tikhonov, A.V. Goncharskii. Moscow : MSU Press, 1987. — 303 p. [in Russian]
  6. Vasil’ev, F.P. Metody optimizatsii / F.P. Vasil’ev. — Moscow : MCCME, 2011. Vol. 1: 620 p.; Vol. 2: 433 p. [in Russian]
  7. Sumin, M.I. Regularized parametric Kuhn–Tucker theorem in a Hilbert space / M.I. Sumin // Comput. Math. Math. Phys. — 2011. — V. 51, № 9. — P. 1489–1509.
  8. Sumin, M.I. On ill-posed problems, extremals of the Tikhonov functional and the regularized Lagrange principles / M.I. Sumin // Russ. Universities Reports. Mathematics. — 2022. — V. 27, № 137. — P. 58–79. [in Russian]
  9. Sumin, V.I. On the iterative regularization of the Lagrange principle in convex optimal control problems for distributed systems of the Volterra type with operator constraints / V.I. Sumin, M.I. Sumin // Differ. Equat. — 2022. — V. 58, № 6. — P. 791–809.
  10. Sumin, V.I. On regularization of the Lagrange principle in the optimization problems for linear distributed Volterra type systems with operator constraints / V.I. Sumin, M.I. Sumin // Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. — 2022. — V. 59. — P. 85–113. [in Russian]
  11. Fursikov, A.V. Optimal Control of Distributed Systems: Theory and Applications / A.V. Fursikov. — Providence : Amer. Math. Soc., 2000. — 305 p.
  12. Troltzsch, F. Optimal Control of Partial Differential Equations. Theory, Methods and Applications /
  13. F. Troltzsch. — Providence; Rhode Island : Amer. Math. Soc., 2010. — 399 p.
  14. Borzi, A. The Sequential Quadratic Hamiltonian Method. Solving Optimal Control Problems / A. Borzi. —
  15. Boca Raton : Chapman and Hall/CRC Press, 2023.
  16. Tonelli, L. Sulle equazioni funzionali di Volterra / L. Tonelli // Bull. Calcutta Math. Soc. — 1929. — V. 20. — P. 31–48.
  17. Tikhonov, A.N. Functional Volterra-type equations and their applications to certain problems of mathematical physics / A.N. Tikhonov // Bull. Mosk. Gos. Univ. Sekt. A. — 1938. — V. 8, № 1. — P. 1–25. [in Russian]
  18. Zabreiko, P.P. Integral Volterra operators / P.P. Zabreiko // Uspekhi Mat. Nauk. — 1967. — V. 22, № 1. —
  19. P. 167–168. [in Russian]
  20. Shragin, I.V. Abstract Nemyckii operators are locally defined operators / I.V. Shragin // Sov. Math. Dokl. — 1976. — V. 17. — P. 354–357.
  21. Sumin, V.I. Volterra functional-operator equations in the theory of optimal control of distributed systems / V.I. Sumin // Sov. Math. Dokl. — 1989. — V. 39, № 2. — P. 374–378.
  22. Zhukovskii, E.S. On the theory of Volterra equations / E.S. Zhukovskii // Differ. Equat. — 1989. — V. 25,
  23. № 9. — P. 1132–1137.
  24. Corduneanu, C. Integral Equations and Applications / C. Corduneanu. — Cambridge; New York : Cambridge
  25. University Press, 1991. — 376 p.
  26. Gohberg, I.C. Theory and Applications of Volterra Operators in Hilbert Space / I.C. Gohberg, M.G. Krein. —
  27. Amer. Math. Soc., 1970. — 378 p.
  28. Bughgeim, A.L. Volterra Equations and Inverse Problems / A.L. Bughgeim. — Utrecht : VSP BV, 1999.
  29. Gusarenko, S.A. On a generalization of the notion of Volterra operator / S.A. Gusarenko // Sov. Math. Dokl. — 1988. — V. 36, № 1. — P. 156–159.
  30. Vath, M. Abstract Volterra equations of the second kind / M. V¨ath // J. Equat. Appl. — 1998. — V. 10, № 9. — P. 125–144.
  31. Zhukovskii, E.S. Abstract Volterra operators / E.S. Zhukovskii, M.J. Alves // Russ. Mathematics. — 2008. —
  32. V. 52, № 3. — P. 1–14.
  33. Sumin, V.I. Functional Volterra equations in the theory of optimal control of distributed systems / V.I. Sumin. — Nizhnii Novgorod : Izd-vo Nizhegorodskogo gosuniversiteta, 1992. — 110 p. [in Russian]
  34. Sumin, V.I. Operators in spaces of measurable functions: The Volterra property and quasinilpotency / V.I. Sumin, A.V. Chernov // Differ. Equat. — 1998. — V. 34, № 10. — P. 1403–1411.
  35. Sumin, V.I. Controlled Volterra functional equations and the contraction mapping principle / V.I. Sumin //
  36. Trudy Inst. Mat. Mekh. UrO RAN. — 2019. — V. 25, № 1. — P. 262–278. [in Russian]
  37. Warga, J. Optimal control of differential and functional equations / J. Warga. — New York : Acad. Press, 1972.
  38. Sumin, M.I. Stable sequential convex programming in a Hilbert space and its application for solving unstable problems / M.I. Sumin // Comput. Math., Math. Phys. — 2014. — V. 54, № 1. — P. 22–44.
  39. Bakushinskii, A.B. Iterative Methods for Solving Ill-Posed Problems / A.B. Bakushinskii, A.V. Goncharskii. — Moscow : Nauka, 1989. — 126 p. [in Russian]
  40. Sumin, M.I. On regularization of the classical optimality conditions in convex optimal control problems /
  41. M.I. Sumin // Trudy Inst. Mat. Mekh. UrO RAN. — 2020. — V. 26, № 2. — P. 252–269. [in Russian]
  42. Sumin, M.I. Nondifferential Kuhn–Tucker theorems in constrained extremum problems via subdifferentials of
  43. nonsmooth analysis / M.I. Sumin // Russ. Universities Reports. Mathematics. — 2020. — V. 25, № 131. —
  44. P. 307–330. [in Russian]
  45. Aubin, J.P. L’analyse non lin/eaire et ses motivations /economiques / J.P. Aubin. — Paris : Masson, 1984. — 214 p.
  46. Loewen, P.D. Optimal Control via Nonsmooth Analysis / P.D. Loewen. — Providence, Rhode Island, USA :
  47. Amer. Math. Soc., 1993. — 153 p.
  48. Jorgens, K. An asymptotic expansion in the theory of neutron transport / K. Jorgens // Comm. Pure Appl.
  49. Math. — 1958. — V. 11, № 2. — P. 219–242.
  50. Morozov, S.F. Non-stationary integro-differential transport equation / S.F. Morozov // Izvestiya vysshikh uchebnykh zavedeniy. Matematika. — 1969. — № 1. — P. 26–31. [in Russian]
  51. Kuznetsov, Yu.A. Correctness of the mixed problem statement for the nonstationary transport equation /
  52. Yu.A. Kuznetsov, S.F. Morozov // Differ. uravneniya. — 1972. — V. 8, № 9. — P. 1639–1648. [in Russian]

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