Numerical computation of a singular-state subarc in an economic optimal control model

In this paper, optimal control problems (OCP) are considered which are characterized by a nonsmooth-state differential equation. More precisely, it is assumed that the right-hand side of the state equation is piecewise smooth and that the junction points between smooth subarcs are determined as roots of a state-dependent switching function. For this kind of OCP necessary conditions are developed. Special attention is paid to the situation that the switching function vanishes identically along a nontrivial subarc. Such subarcs, which are called singular-state subarcs, are investigated with respect to the necessary conditions and to the junction conditions. In this paper, we assume that the switching function is of first order with respect to the control. The theory is applied to an economic optimal control model due to Pohmer, which describes the personal income distribution of a typical consumer, who wants to maximize the total utility of his lifetime by controlling the consumption, the rate of the total time used for working, and the rate of working time used for education and extended professional training. The state variables are the human capital and the capital itself. The utility function contains different parts which represent the influence of consumption, time of recreation, and human capital. Into this problem a parameter enters which describes the interest rate of capital. It is obvious that this parameter in general will differ for positive and negative values of the capital. Thus, the resulting OCP in a natural way becomes a nonsmooth one. For this problem, the necessary conditions are derived and numerical solutions are presented which are obtained by an indirect optimal control method. It turns out that for a certain distance of the positive and negative interest rate, the optimal solution contains a singular-state subarc.