Optimization MethodsVariational problems which are interesting from physical and technical viewpoints are often supplemented with ordinary differential equations as constraints, e. g., in the form of Newton's equations of motion. Since analytical solutions for such problems are possible only in exceptional cases and numerical treat ment of extensive systems of differential equations formerly caused computational difficulties, in the classical calculus of variations these problems have generally been considered only with respect to their theoretical aspects. However, the advent of digital computer installations has enabled us, approximately since 1950, to make more practical use of the formulas provided by the calculus of variations, and also to proceed from relationships which are oriented more numerically than analytically. This has proved very fruitful since there are areas, in particular, in automatic control and space flight technology, where occasionally even relatively small optimization gains are of interest. Further on, if in a problem we have a free function of time which we may choose as advantageously as possible, then determination of the absolutely optimal course of this function appears always advisable, even if it gives only small improve ments or if it leads to technical difficulties, since: i) we must in any case choose some course for free functions; a criterion which gives an optimal course for that is very practical ii) also, when choosing a certain technically advantageous course we mostly want to know to which extent the performance of the system can further be increased by variation of the free function. |
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according angle approximation arbitrary assume Bellman method boundary conditions boundary values calculate calculus of variations computation consider const constant control functions corresponding dependence determined du(t end point equations as constraints equations of motion Euler equation example extreme value final conditions final values flight follows given gradient method grid points Hamilton-Jacobi Hamiltonian height of climb hence initial values integral interval iteration Lagrange multipliers Lagrange problem linear m/sec m₁ massflow Mayer problem Miele minimum necessary conditions Newton-Raphson method numerical obtain optimization problem optimum orbit ordinary differential equations partial differential equation path plane Pontryagin maximum principle prescribed procedure recursive formula relationship respect satisfied solution solve sounding rocket step stop condition substitute t₁ tangent theorem trajectory u₁ value problem variables velocity xj(t y₁ δυ λο Σ λ Σ Σ ӘР ди ду