Dynamic optimal control of process–product innovation with learning by doing

• Presenting a model for the process–product innovation with learning by doing.
• Investigating the relationships among product and process innovation.
• Investigating the extension model's optimal conditions and characteristics.

In this paper, we present a dynamic optimal control model of process–product innovation with learning by doing, and extend the model of Chenavaz (2012) to an even more general model in which the firm's cost functions of product and process innovation depend on both the innovation investments and the knowledge accumulations of product and process innovation; furthermore, in our paper, the product price, the investments of product and process innovation are decision variables; the product quality, production cost, the change rates of knowledge accumulations of product and process innovation are state variables. The main objective of this paper is to analyze the relationships between these variables, and investigate the model's optimal conditions and characteristics. Further, we solve the model with some numerical examples, and sensitivity analysis is conducted to study the effect of changing the parameters and coefficients on the objective function value.