On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function
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 [\textit{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 \textit{F. P. Ramsey} [``A mathematical theory of saving'', Econ. J. 38, No. 152, 543--559 (1928), \url{http://www.jstor.org/stable/2224098}].
scientific article; zbMATH DE number 6381004
scientific article