Mathematical Notes

, Volume 103, Issue 1–2, pp 167–174 | Cite as

On Balder’s Existence Theorem for Infinite-Horizon Optimal Control Problems

  • K. O. Besov


Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f0, 0} of the utility function (integrand) f0 is relaxed to the requirement that the integrals of f0 over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.


optimal control existence theorem infinite horizon 


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  1. 1.
    E. J. Balder, “An existence result for optimal economic growth problems,” J. Math. Anal. Appl. 95 (1), 195–213 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. V. Dmitruk and N. V. Kuz’kina, “Existence theorem in the optimal control problem on an infinite time interval,” Mat. Zametki 78 (4), 503–518 (2005) [Math. Notes 78 (3–4), 644–480 (2005)]; “Erratum,” Mat. Zametki 80 (2), 320 (2006) [Math. Notes 80 (1–2), 309 (2006)].MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    E. J. Balder, “Lower semicontinuity of integral functionals with nonconvex integrands by relaxation-compactification,” SIAM J. Control Optim. 19 (4), 533–542 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Bogusz, “On the existence of a classical optimal solution and of an almost strongly optimal solution for an infinite-horizon control problem,” J. Optim. Theory Appl. 156 (3), 650–682 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. M. Aseev, “Existence of an optimal control in infinite horizon problems with an unbounded set of constraints on the control,” in Tr. Inst. Mat. Mekh. (Ural Branch, Russian Academy of Sciences) (2016), Vol. 22, No. 2, pp. 18–27, [Proc. Steklov Inst. Math. 297, Suppl. 1, 1–10 (2017)].Google Scholar
  6. 6.
    V. Lykina, “An existence theorem for a class of infinite horizon optimal control problems,” J. Optim. Theory Appl. 169 (1), 50–73 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. M. Aseev, K. O. Besov, and A. V. Kryazhimskii, “Infinite-horizon optimal control problems in economics,” UspekhiMat. Nauk 67 (2 (404)), 3–64 (2012) [Russian Math. Surveys 67 (2), 195–253 (2012)].MathSciNetCrossRefGoogle Scholar
  8. 8.
    S. M. Aseev, “Adjoint variables and intertemporal prices in infinite-horizon optimal control problems,” in Trudy Mat. Inst. Steklova, Vol. 290: Contemporary Problems of Mathematics, Mechanics, and Mathematical Physics (MAIK, Moscow, 2015), pp. 239–253 [Proc. Steklov Inst. Math. 290, (1), 223–237 (2015)].Google Scholar
  9. 9.
    S. M. Aseev, “Optimization of dynamics of controlled systemwith risk factors,” in Tr. Inst. Mat. Mekh. (Ural Branch Russian Academy of Sciences) (2017), Vol. 23, No. 1, pp. 27–42, [in Russian].MathSciNetGoogle Scholar
  10. 10.
    K. O. Besov, “On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function,” in Trudy Mat. Inst. Steklova, Vol. 284: Functional Spaces and Related Problems of Analysis (MAIK, Moscow, 2014), pp. 56–88 [Proc. Steklov Inst. Math. 284, (1), 50–80 (2017)].Google Scholar
  11. 11.
    K. O. Besov, “Problem of optimal endogenous growth with exhaustible resources and possibility of a technological jump,” in Trudy Mat. Inst. Steklova, Vol. 291: Optimal Control (MAIK, Moscow, 2015), pp. 56–68 [Proc. Steklov Inst. Math. 291, (1), pp. 49–60 (2017)].Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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