The shortest time and optimal investment control in duopoly

Abstract

In this paper, we consider an optimal duopoly competition problem in commencing period of horizontal expansion. First, we propose a type of optimal duopoly competition problem including the shortest time and the minimal investment cost to reach a balance for one product from two parties in terms of product occupancy rate. Such problem formulation includes a nonlinear differential-algebraic model with continuous inequality constraints and terminal state constraints. Due to the presence of the continuous state inequality constraints, maybe there are infinite constraints to be satisfied on the time horizon. Thus, it is challenging to obtain a feasible advertising strategy. In order to solve this problem effectively, we apply a constraint transcription technique to covert these constraints into some canonical forms. The converted problem is then solved via the control parameterization method. The essential idea of this method is to approximate the control function with a piecewise constant function in such a way that the optimal control problem is transformed into an optimal parameter selection problem, which can be solved as a nonlinear programming problem. Particularly, in order to optimize the switching time for reducing the investment cost, a time scaling transform is developed. Finally, a beer sales problem is studied to show the effectiveness of the proposed method, and we also compare the investment strategies with applying the time scaling transform and the coarsely equal partition over the time horizon.