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On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function. (English) Zbl 1310.91101
Proc. Steklov Inst. Math. 284, 50-80 (2014) and Tr. Mat. Inst. Steklova 284, 56-88 (2014).
The paper investigates the optimal control infinite-horizon problem of Pontryagin type (problem (P)):
\[ \begin{aligned} & \dot {x}(t) = f(x(t),u(t)) \in \mathbb{R}^{n}, \qquad u(t)\in U \subset \mathbb{R}^{m}, \\ & x(0)= x_{0} \in G \quad \text{(an open set)}, \\ & J(x(\cdot), u(\cdot)) = \int_{0}^{\infty} {e^{ - \rho t}g} \left( {x(t), u(t)} \right) dt \to \max. \end{aligned} \] It extends the results and methods of a previous paper by the author in collaboration [S. M. Aseev et al., Russ. Math. Surv. 67, No. 2, 195–253 (2012); translation from Usp. Mat. Nauk 67, No. 2, 3–64 (2012; Zbl 1248.49023)]. Novelties of the present paper are: the admissible control set \(U\) is a bounded set ‘which is open in its closure (i.e. \(\bar {U}\backslash U\) is closed)’, and ‘the instantaneous utility function’ \(g\left( {x,u} \right) \to - \infty \) whenever \(\left( {x,u} \right) \to \left( {\bar {x},\bar {u}} \right) \in G \times \left( {\bar {U}\backslash U} \right)\). The author obtains (under some analytical assumptions) main qualitative results with respect to this variation of the optimal control problem: a solution existence to problem (P), an explicit formula for the adjoint function, and a variant of the Pontryagin maximum principle via approximation of the problem by a sequence of finite-horizon ones. These results are demonstrated on the one-sector model of the economic growth of F. P. Ramsey [“A mathematical theory of saving”, Econ. J. 38, No. 152, 543–559 (1928), http://www.jstor.org/stable/2224098].

MSC:
91B62 Economic growth models
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
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