The Pontryagin maximum principle and transversality conditions for an optimal control problem with infinite time interval (Q2784570)

The following optimal control problem that arises in mathematical economics is considered: \(J(x,u)=\int_0^{\infty} \exp (-\rho t) \ln D(x,u) dt \to \max\) subject to \({\dot x}=f(x,u), u \in U\), \(x(0)=x_0\), where \(x=(x^1, \dots, x^n), u=(u^1, \dots, u^m)\), \(U\) is a nonempty compact subset of \(\mathbb{R}^m\), and \(\rho >0\). The maximum is sought for the class of measurable functions \(u: [0, \infty) \to \mathbb{R}^m\) bounded on each finite time interval \([0,T]\). The aim of the work is to obtain first-order necessary optimality conditions for this problem. The basic method used in the paper consists in the approximation of the original optimal control problem on an infinite time interval by a sequence of classical problems posed on finite intervals. Necessary conditions for the original problem are obtained by passage to the limit in the relations of Pontryagin maximum principle for the approximation problems.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00023].