Dynamic Programming and the Calculus of Variations
A demonstration of the relationships between the calculus of variations, a mathematical discipline concerning certain problems of optimization theory, and dynamic programming, a newer mathematical approach applicable to optimization problems. In addition to explaining and contrasting the two approaches, the Report shows that many results of the calculus of variations become simple and intuitively apparent when examined from the dynamic programming viewpoint. In emphasizing the geometrical and physical insight afforded by this approach, the study shows how these techniques can be applied, for instance tostochastic and adaptive variational problems. It can be used in the study of dynamic programming and other new mathematical formalisms; in optimal control problems, such as the determination of rocket trajectories, the correction of launch errors and inflight disturbances of spacecraft; and in the problems of optimal control found in economics, biology, and the social sciences. (Author).
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An Example of a Multistage Decision Process Problem
The Dynamic Programming Solution of the Example
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absolute minimality admissible curves arc-length associated boundary condition calculus of variations candidate curve Chapter classical consider control function criterion function decision deduced definite integral determined dynamic programming end point equals zero Euler curve Euler-Lagrange equation evaluated expected value function y(x fundamental equation fundamental partial differential given inequality constraint initial point Jacobi condition Legendre condition minimizing curve minimum value multiplier functions multiplier rule necessary condition obtain optimal control optimal curve optimal policy function optimal trajectory optimal value function ordinary differential equations partial differential equation particular path perturbed point x0 problem discontinuity properties recurrence relation region relative minimality result satisfied second partial derivatives Section simplest problem solution curve specified stochastic strong relative minimum terminal conditions terminal control problem terminal manifold terminal point terminal value tion transversality conditions variable variational problem vector vertex weak neighborhood weak relative minimum Weierstrass