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subject to the dynamic constraint _x = u, as well as the initial condition x(0) = x 0 and the terminal condition allowing x(T) to be chosen freely. The associated Hamiltonian is H = x2 cu2 + pu with a minus sign to convert the minimization problem into a maximization problem. The associated extended Hamiltonian is H~ = x2 cu2 + pu + _px
now considered to be Dynamic Optimization. Several adaptations of the theory were later required, including extensions to stochastic models and in nite dimensional processes. These lecture notes are intended as a friendly introduction to Calculus of Variations and Optimal Control, for students in science, engineering and economics with a general
A more extensive analysis of dynamic optimisation can be found in the appendixes of the following books: A. Mas-Colell, M.D. Whinston and J.R. Green: Microeconomic Theory, Oxford Univer-sity Press (1995). M. Wickens: Macroeconomic theory: A dynamic General Equilibrium Approach, Prince-ton University Press (2009) 3
This equation tells us that solving the constrained optimization problem requires that kT+1 has to be set equal to zero unless λT is equal to zero, that is, unless the economic agent is completely satiated with consumption. The transversality condition can be obtained by taking the limit of 1.9a as T ∞.
The calculus of variations is used to optimize afunctional that maps functions into real numbers. A typical problem is to choose a function [t 0;t 1] 3t 7!x(t) 2R, often denoted simply by x, in order to maximize the integralobjective functional J(x) = Z t 1 t0 F(t;x(t);x_(t))dt subject to the xed end point conditions x(t 0) = x 0, x(t 1) = x 1.
This course focuses on dynamic optimization methods, both in discrete and in continuous time. We approach these problems from a dynamic programming and optimal control perspective. We also study the dynamic systems that come from the solutions to these problems. The course will illustrate how these techniques are useful in various applications ...
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