State-Constrained Optimal Control of a Coupled Quasilinear Parabolic System Modeling Economic Growth in the Presence of Technological Progress

Abstract

We develop an optimal control framework that enables to determine the most-beneficial ways of investing in technology and directing capital within an economy. Our developed framework features three main novelties: the optimization of a cross–diffusion term that incorporates the allocation of capital towards specific regions with higher level of technology; the coupling of technological progress with the capital in the state system; and the inclusion of an inequality constraint imposing that the squared norm of technological progress does not surpass a capacity \(M_A>0\), which is more practical in economic applications. This leads to a new state-constrained optimal control problem which we analyze as follows. First, by examining the weak well-posedness of the dynamics, we identify a threshold parameter \(M^*>0\) such that when \(M_A\ge M^*\), the state-constraint can be omitted. In this case, we deal with a reduced state-unconstrained optimal control problem. On the other hand, when \(M_A<M^*\), the state-constraint is not implicitly incorporated. Consequently, we proceed by a penalization approach to formulate a sequence of state-unconstrained optimal control problems and provide necessary optimality conditions for its associated sequence of locally optimal solutions. Subsequently, we prove that the sequence of locally optimal solutions converges strongly to a locally optimal solution for the original state-constrained optimal control problem and retrieve its necessary optimality conditions. Finally, we perform various numerical simulations to illustrate the effects of optimal investment in technology and optimal capital direction on the economy. This study could offer interesting insights in the perspective of circular economy transition.