Another view of the maximum principle for infinite-horizon optimal control problems in economics

S. M. Aseev ; V. M. Veliov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 6, pp. 963-1011

Résumé

The authors present their recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The main distinguishing feature of the result is that the adjoint variable is explicitly specified by a formula analogous to the Cauchy formula for solutions of linear differential systems. In certain situations this formula implies the ‘standard’ transversality conditions at infinity. Moreover, it can serve as an alternative to them. Examples demonstrate the advantages of the proposed version of the maximum principle. In particular, its applications are considered to Halkin's example, to Ramsey's optimal economic growth model, and to a basic model for optimal extraction of a non-renewable resource. Also presented is an economic interpretation of the characterization obtained for the adjoint variable. Bibliography: 62 titles.

Détail

Keywords: optimal control, Pontryagin maximum principle, transversality conditions, Ramsey model, optimal extraction of a non-renewable resource.
Mots-clés : adjoint variables

Bibliographie

[1] D. Acemoglu, Introduction to modern economic growth, Princeton Univ. Press, Princeton, NJ, 2009, xviii+990 pp. | Zbl

[2] V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin, Optimal control, Contemp. Soviet Math., Consultants Bureau, New York, 1987, xiv+309 pp. | DOI | MR | MR | Zbl | Zbl

[3] K. J. Arrow, M. Kurz, Public investment, the rate of return, and optimal fiscal policy, J. Hopkins Univ. Press, Baltimore, MD, 1970, 218 pp.

[4] S. M. Aseev, “On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems”, Proc. Steklov Inst. Math. (Suppl.), 287, suppl. 1 (2014), 11–21 | DOI | MR | Zbl

[5] S. M. Aseev, “Adjoint variables and intertemporal prices in infinite-horizon optimal control problems”, Proc. Steklov Inst. Math., 290:1 (2015), 223–237 | DOI | DOI | MR | Zbl

[6] S. M. Aseev, K. O. Besov, S. Yu. Kaniovski, “Optimal policies in the Dasgupta–Heal–Solow–Stiglitz model under nonconstant returns to scale”, Proc. Steklov Inst. Math., 304 (2019), 74–109 | DOI | DOI | MR | Zbl

[7] S. M. Aseev, K. O. Besov, A. V. Kryazhimskiy, “Infinite-horizon optimal control problems in economics”, Russian Math. Surveys, 67:2 (2012), 195–253 | DOI | DOI | MR | Zbl

[8] S. M. Aseev, M. I. Krastanov, V. M. Veliov, “Optimality conditions for discrete-time optimal control on infinite horizon”, Pure Appl. Funct. Anal., 2:3 (2017), 395–409 | MR | Zbl

[9] S. M. Aseev, A. V. Kryazhimskiĭ, “The Pontryagin maximum principle for an optimal control problem with a functional specified by an improper integral”, Dokl. Math., 69:1 (2004), 89–91 | MR | Zbl

[10] S. M. Aseev, A. V. Kryazhimskiy, “The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons”, SIAM J. Control Optim., 43:3 (2004), 1094–1119 | DOI | MR | Zbl

[11] S. M. Aseev, A. V. Kryazhimskii, “The Pontryagin maximum principle and optimal economic growth problems”, Proc. Steklov Inst. Math., 257 (2007), 1–255 | DOI | MR | Zbl

[12] S. M. Aseev, A. V. Kryazhimskii, “On a class of optimal control problems arising in mathematical economics”, Proc. Steklov Inst. Math., 262 (2008), 10–25 | DOI | MR | Zbl

[13] S. M. Aseev, A. V. Kryazhimskii, A. M. Tarasyev, “The Pontryagin maximum principle and transversality conditions for an optimal control problem with infinite time interval”, Proc. Steklov Inst. Math., 233 (2001), 64–80 | MR | Zbl

[14] S. M. Aseev, V. M. Veliov, “Maximum principle for infinite-horizon optimal control problems with dominating discount”, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19:1-2 (2012), 43–63 | MR | Zbl

[15] S. M. Aseev, V. M. Veliov, “Needle variations in infinite-horizon optimal control”, Variational and optimal control problems on unbounded domains, Contemp. Math., 619, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 2014, 1–17 | DOI | MR | Zbl

[16] S. M. Aseev, V. M. Veliov, “Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions”, Tr. IMM UrO RAN, 20, no. 3, 2014, 41–57 ; Proc. Steklov Inst. Math. (Suppl.), 291, suppl. 1 (2015), 22–39 | MR | Zbl | DOI

[17] J. P. Aubin, F. H. Clarke, “Shadow prices and duality for a class of optimal control problems”, SIAM J. Control Optim., 17:5 (1979), 567–586 | DOI | MR | Zbl

[18] E. J. Balder, “An existence result for optimal economic growth problems”, J. Math. Anal. Appl., 95:1 (1983), 195–213 | DOI | MR | Zbl

[19] R. J. Barro, X. Sala-i-Martin, Economic growth, McGraw Hill, New York, 1995, xviii+539 pp.

[20] R. Bellman, Dynamic programming, Princeton Univ. Press, Princeton, NJ, 1957, xxv+342 pp. | MR | MR | Zbl

[21] H. Benchekroun, C. Withagen, “The optimal depletion of exhaustible resources: a complete characterization”, Resource and Energy Economics, 33:3 (2011), 612–636 | DOI

[22] L. M. Benveniste, J. A. Scheinkman, “Duality theory for dynamic optimization models of economics: the continuous time case”, J. Econom. Theory, 27:1 (1982), 1–19 | DOI | MR | Zbl

[23] K. O. Besov, “On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function”, Proc. Steklov Inst. Math., 284 (2014), 50–80 | DOI | DOI | MR | Zbl

[24] Yu. I. Brodskii, “Necessary conditions for a weak extremum in optimal control problems on an infinite time interval”, Math. USSR-Sb., 34:3 (1978), 327–343 | DOI | MR | Zbl

[25] P. Cannarsa, H. Frankowska, “Value function, relaxation, and transversality conditions in infinite horizon optimal control”, J. Math. Anal. Appl., 457:2 (2018), 1188–1217 | DOI | MR | Zbl

[26] D. A. Carlson, A. B. Haurie, A. Leizarowitz, Infinite horizon optimal control. Deterministic and stochastic systems, 2nd rev. and enl. ed., Springer-Verlag, Berlin, 1991, xvi+332 pp. | DOI | MR | Zbl

[27] D. Cass, “Optimum growth in an aggregative model of capital accumulation”, Rev. Econom. Stud., 32:3 (1965), 233–240 | DOI

[28] L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Ergeb. Math. Grenzgeb. (N. F.), 16, Springer-Verlag, Berlin–Göttingen– Heidelberg, 1959, vii+271 pp. | DOI | MR | Zbl

[29] A. C. Chiang, Elements of dynamic optimization, MacGraw-Hill, Singapore, 1992, xiii+327 pp.

[30] F. H. Clarke, Optimization and nonsmooth analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Intersci. Publ. John Wiley Sons, Inc., New York, 1983, xiii+308 pp. | MR | MR | Zbl | Zbl

[31] F. Clarke, Functional analysis, calculus of variations and optimal control, Grad. Texts in Math., 264, Springer, London, 2013, xiv+591 pp. | DOI | MR | Zbl

[32] P. Dasgupta, G. Heal, “The optimal depletion of exhaustible resources”, Rev. Econom. Stud., 41:5 (1974), 3–28 | DOI | Zbl

[33] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967, 472 pp. | MR | Zbl

[34] R. Dorfman, “An economic interpretation of optimal control theory”, Amer. Econom. Rev., 59:5 (1969), 817–831

[35] I. Ekeland, “Some variational problems arising from mathematical economics”, Mathematical economics (Montecatini Terme, 1986), Lecture Notes in Math., 1330, Springer, Berlin, 1988, 1–18 | DOI | MR | Zbl

[36] A. F. Filippov, Differential equations with discontinuous righthand sides, Math. Appl. (Soviet Ser.), 18, Kluwer Acad. Publ., Dordrecht, 1988, x+304 pp. | DOI | MR | MR | Zbl | Zbl

[37] H. Halkin, “Necessary conditions for optimal control problems with infinite horizons”, Econometrica, 42:2 (1974), 267–272 | DOI | MR | Zbl

[38] P. Hartman, Ordinary differential equations, John Wiley Sons, Inc., New York–London–Sydney, 1964, xiv+612 pp. | MR | MR | Zbl | Zbl

[39] H. Hotelling, “The economics of exhaustible resources”, J. Polit. Econom., 39:2 (1931), 137–175 | DOI | Zbl

[40] S. Hu, N. S. Papageorgiou, Handbook of multivalued analysis, v. I, Math. Appl., 419, Theory, Kluwer Acad. Publ., Dordrecht, 1997, xvi+964 pp. | MR | Zbl

[41] A. D. Ioffe, V. M. Tihomirov, Theory of extremal problems, Stud. Math. Appl., 6, North-Holland Publ. Co., Amsterdam–New York, 1979, xii+460 pp. | MR | MR | Zbl | Zbl

[42] T. Kamihigashi, “Necessity of transversality conditions for infinite horizon problems”, Econometrica, 69:4 (2001), 995–1012 | DOI | MR | Zbl

[43] T. C. Koopmans, “On the concept of optimal economic growth”, The econometric approach to development planning, North Holland Publ. Co., Amsterdam; Rand McNally, Chicago, 1965

[44] P. Michel, “On the transversality condition in infinite horizon optimal problems”, Econometrica, 50:4 (1982), 975–985 | DOI | MR | Zbl

[45] S. Pickenhain, “Hilbert space treatment of optimal control problems with infinite horizon”, Modeling, simulation and optimization of complex processes – 2012, Springer, Cham, 2014, 169–182 | DOI

[46] S. Pickenhain, “Infinite horizon optimal control problems in the light of convex analysis in Hilbert spaces”, Set-Valued Var. Anal., 23:1 (2015), 169–189 | DOI | MR | Zbl

[47] L. S. Pontryagin, “Nekotorye matematicheskie zadachi, voznikayuschie v svyazi s teoriei optimalnykh sistem avtomaticheskogo regulirovaniya”, Sessiya AN SSSR po nauchnym problemam avtomatizatsii proizvodstva (15–20.X 1956 g.), v. 2, Osnovnye problemy avtomaticheskogo regulirovaniya i upravleniya, Izd-vo AN SSSR, M., 1957, 107–117

[48] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Intersci. Publ. John Wiley Sons, Inc., New York–London, 1962, viii+360 pp. | MR | MR | Zbl | Zbl

[49] F. P. Ramsey, “A mathematical theory of saving”, Econom. J., 38:152 (1928), 543–559 | DOI

[50] N. Sagara, “Value functions and transversality conditions for infinite-horizon optimal control problems”, Set-Valued Var. Anal., 18:1 (2010), 1–28 | DOI | MR | Zbl

[51] A. Seierstad, “Necessary conditions for nonsmooth, infinite-horizon optimal control problems”, J. Optim. Theory Appl., 103:1 (1999), 201–229 | DOI | MR | Zbl

[52] A. Seierstad, “A maximum principle for smooth infinite horizon optimal control problems with state constraints and with terminal constraints at infinity”, Open J. Optim., 4:3 (2015), 100–130 | DOI

[53] A. Seierstad, K. Sydsæ{ter}, “Sufficient conditions in optimal control theory”, Internat. Econom. Rev., 18:2 (1977), 367–391 | DOI | MR | Zbl

[54] A. Seierstad, K. Sydsæ{ter}, Optimal control theory with economic applications, Adv. Textbooks Econom., North-Holland Publ. Co., Amsterdam, 1987, xvi+445 pp. | MR | Zbl

[55] A. Seierstad, K. Sydsæ{ter}, “Conditions implying the vanishing of the Hamiltonian at infinity in optimal control problems”, Optim. Lett., 3:4 (2009), 507–512 | DOI | MR | Zbl

[56] K. Shell, “Applications of Pontryagin's maximum principle to economics”, Mathematical systems theory and economics (Varenna, 1967), Lect. Notes Oper. Res. Math. Econom., I, Springer, Berlin, 1969, 241–292 | MR | Zbl

[57] B. Skritek, V. M. Veliov, “On the infinite-horizon optimal control of age-structured systems”, J. Optim. Theory Appl., 167:1 (2015), 243–271 | DOI | MR | Zbl

[58] G. V. Smirnov, “Transversality condition for infinite-horizon problems”, J. Optim. Theory Appl., 88:3 (1996), 671–688 | DOI | MR | Zbl

[59] R. M. Solow, “Intergenerational equity and exhaustible resources”, Rev. Econom. Stud., 41:5 (1974), 29–45 | DOI | Zbl

[60] J. Stiglitz, “Growth with exhaustible natural resources: efficient and optimal growth paths”, Rev. Econom. Stud., 41:5 (1974), 123–137 | DOI | Zbl

[61] N. Tauchnitz, “The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon”, J. Optim. Theory Appl., 167:1 (2015), 27–48 | DOI | MR | Zbl

[62] J. J. Ye, “Nonsmooth maximum principle for infinite-horizon problems”, J. Optim. Theory Appl., 76:3 (1993), 485–500 | DOI | MR | Zbl