- Dynamical systems. Differential equations and finite differences equations. Systems of differential equations and finite differences equations. Equilibrium solutions for dynamical systems and stability of solutions.The linear case: solutions and stability of equilibrium solutions. Nonlinear case: linearization and Liapunov method.
- Dynamic optimization. Calculus of variation and Eulero’s equation. Transversality conditions. Sufficient optimality conditions. Optimal control and Maximum Principle. Transversality conditions. The case with infinite horizon. Autonomous problems. Economical applications.
- Dynamic programming. Dynamic optimization and Bellman’s principle. Economical applications.
The course introduces the most important instruments needed to analyze and study the behaviour and properties of solutions of a dynamical system. The problem of Dynamic Optimization is then introduced both in the continuous and the discrete case, and necessary and sufficient optimality conditions are given. Finally, some economical applications of such instruments are presented.
The contents of this course are fundamental for many economical models.