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

## Abstract

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{*f*_{0}, 0} of the utility function (integrand) *f*_{0} is relaxed to the requirement that the integrals of *f*_{0} 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.

## Keywords

optimal control existence theorem infinite horizon## Preview

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## References

- 1.E. J. Balder, “An existence result for optimal economic growth problems,” J. Math. Anal. Appl.
**95**(1), 195–213 (1983).MathSciNetCrossRefMATHGoogle Scholar - 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)].MathSciNetCrossRefMATHGoogle Scholar - 3.E. J. Balder, “Lower semicontinuity of integral functionals with nonconvex integrands by relaxation-compactification,” SIAM J. Control Optim.
**19**(4), 533–542 (1981).MathSciNetCrossRefMATHGoogle Scholar - 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).MathSciNetCrossRefMATHGoogle Scholar - 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.V. Lykina, “An existence theorem for a class of infinite horizon optimal control problems,” J. Optim. Theory Appl.
**169**(1), 50–73 (2016).MathSciNetCrossRefMATHGoogle Scholar - 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.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.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.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.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