The Pontryagin maximum principle for an optimal control problem with a functional defined by an improper integral

A normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable is proved for a class of infinite-horizon optimal control problems motivated by economic applications. The result is similar to the variant of the maximum principle for problems with dominating discount that was developed earlier in [1], [2] with the use of the method of finite-horizon approximations. The main novelty of the result is that it is proved under relaxed assumptions on the convergence of the integral utility functional. In particular, the result can be applied also for problems where the objective value may be infinite. In this case the notion of optimality of admissible control is specified in a special way. The proof of the result is based on application of classical needle variations technique.
It should be noted that earlier the needle variation technique was applied to developing the maximum principle for infinite-horizon problems in monograph “The mathematical theory of optimal processes” by L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze, and E. F. Mishchenko (Fizmatgiz, Moscow, 1961). Nevertheless, the direct application of this technique does not provide any additional characterizations of adjoint variables (such as normality of the problem and/or validity of some additional boundary conditions at infinity) although these issues play a central role in problems of this type.
The result is obtained in collaboration with Prof. V. M. Veliov (Vienna University of Technology, Austria) and published in [3], [4].

References

1. S. M. Aseev, A. V. Kryazhimskii, “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        
2. S. M. Aseev, A. V. Kryazhimskii, “The Pontryagin maximum principle and optimal economic growth problems”, Proc. Steklov Inst. Math., 257 (2007), 1–255            
3. 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      
4. S. M. Aseev, V. M. Veliov, Needle variations in infinite-horizon optimal control, Research Report 2012-04, ORCOS, Vienna University of Technology, Vienna, 2012, 21 pp. 2012-04_Ase-VV_new.pdf