The analysis of the dynamic optimization problem in econophysics from the point of view of the symplectic approach for constrained systems

A standard approach to deal with the problems relative to dynamic optimization is the well known Pontryagin method to obtain a concise condition for an optimal control, minimizing the cost functional. In this paper we have proposed a simpler dynamic optimization procedure through a so-called symplectic algorithm. We worked with the analogy between the physical systems at the classical and quantum energy levels. In this way, we started by considering some cost functional $f(q,u,t),$ where $q$ and $u$ are state and control variables, respectively. We have shown that it is possible to investigate the system by means of a symplectic extension, where we can reduce any constrained system into its canonical first order form. Consequently, the dynamics of evolution, usually governed by Dirac's constraint method, was re-obtained here in a very elegant way. On the other hand, the constraints classification turned out to be different from the one used in Dirac's procedure. Of course, our analysis is valid for unconstrained systems where the Pontryagin equations are valid too. Four optimization problems were solved.