Variational 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.
Reihe
Auflage
Softcover reprint of the original 1st ed. 1975
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
Maße
Höhe: 28 cm
Breite: 21.6 cm
Gewicht
ISBN-13
978-3-540-07194-5 (9783540071945)
DOI
10.1007/978-3-642-87731-5
Schweitzer Klassifikation
I. Basic Concepts.- 1. Review of Methods to be Discussed and their Interrelations.- 1.1. Statement of Problem.- 1.2. Characterization of Various Optimization Methods.- 1.3. Schematic Interrelations and Historical Order of Methods.- 2. A General Outline of the Calculus of Variations.- 2.1. Explanation of the Basic Concepts via the Approach of Caratheodory.- 2.1.1. The Approach of Caratheodory.- 2.1.2. The Euler and Hamilton-Jacobi Differential Equation.- 2.1.3. Transversality.- 2.1.4. Regularity.- 2.1.5. Calculations of Extremals from Curves of Equal Extreme Value and Conversely.- 2.1.6. Example for dealing with a Minimal Problem by the Euler and Hamilton-Jacobi Differential Equation.- 2.1.7. The Erdmann-Weierstrass Corner Conditions.- 2.2. Theory of Integrands Linear in y?.- 2.2.1. Statement of Problem.- 2.2.2. Establishment of the Character of the Extremal when it is not dependent on y?.- 2.2.3. The Miele Problem Formulation.- 2.2.4. The Maximum Height of Climb of a Sounding Rocket and the Miele Theory.- 2.3. Variational Problems with Differential Equations as Constraints.- 2.3.1. Generalizations of the Basic Concepts of the Calculus of Variations.- 2.3.2. Peculiarities of the Differential Equations as Constraints.- 2.3.3. Lagrange, Meyer and Bolza Problem.- II. Indirect Methods.- 1. The Pontryagin Maximum Principle.- 1.1. The Fundamental Theorem.- 1.1.1. Statement of Problem.- 1.1.2. Necessary Conditions for 1.1.1..- 1.1.3. Additional Comments.- 1.2. The Theorems of the Pontryagin Theory.- 1.2.1. Basic Ideas.- 1.2.2. The Major Theorems of the Pontryagin Theory.- 1.2.3. Treatment of the Maximum Climb of a Sounding Rocket by means of the Pontryagin Theory.- 1.2.4. Singular Arcs.- 1.3. Linear Time-Optimal Systems.- 1.3.1. Peculiarities of Linear Systems.- 1.3.2. Two Characteristic Examples.- 1.3.3. General Relationships.- 1.4. The Synthesis Problem.- 1.4.1. Problem Formulation.- 1.4.2. General Statements on the Synthesis Problem.- 2. Adjustment of the Calculus of Variations to the Recent Problem Formulations.- 2.1. The Mayer and Lagrange Problems with the Pontryagin Distinction between State Variables and Control Functions.- 2.1.1. Re-formulation of the Mayer and Lagrange Problems.- 2.1.2. Comparison of Resulting Conditions with Pontryagin Max. Principle.- 2.1.3. The Principle of Simplifying the Problem by Extending the Constraints.- 2.1.4. Example: Optimal Flight in Vacuum.- 2.2. Simple Derivation of the Requirements induced by Constraints for the Existence of an Optimum.- 2.2.1. The Lagrange Derivation of the Euler Equation.- 2.2.2. Application of the Lagrange Derivation to more general Problems by Formal Extension.- 2.2.3. Simple Constraints for the Control Functions.- 2.3. General Treatment of Inequalities as Constraints.- 2.3.1. Formulation of the Constraint by Inequalities.- 2.3.2. Optimization Conditions in the Presence of Inequalities as Constraints.- 2.3.3. An Identical Solution of the Euler Equations for the Constrained Part.- 2.3.4. An Example for Optimization Problems with Inequalities as Constraints.- 2.4. Jumps in the State Variables.- 2.4.1. Statement of Problem.- 2.4.2. Conditions for Jumps in the State Variables.- 3. Numerical Solution of the Boundary Value Problem for Systems of Ordinary Non-Linear Differential Equations.- 3.1. Basic Concepts.- 3.1.1. Problem Formulation.- 3.1.2. Basic Equations.- 3.2. Iterative Fulfilment of the Boundary Conditions while satisfying the Differential Equations.- 3.2.1. Systematic Variation of the Initial Values.- 3.2.2. Example: Optimal Flight in Vacuum.- 3.2.3. Exact Calculation of the Partial Derivatives with respect to the Free Initial Values.- 3.2.4. Further Methods.- 3.3. Iterative Fulfilment of the Differential Equations.- 3.3.1. The Basic Principle.- 3.3.2. Similarity of the Method with the Newton-Raphson Method for Determining the Roots of a Function f(z) = 0.- 3.3.3. Additional Remarks.- III. Direct Methods.- 1. Gradient Method of the First Order.- 1.1. The Gradient Method for Ordinary Extremal Problems.- 1.1.1. Basic Equations.- 1.1.2. Rocket Staging Optimization as an Example of the Gradient Method.- 1.1.3. Ordinary Extremal Problems with Constraints.- 1.1.4. Limits of the Gradient Method.- 1.2. The Gradient Method for Simple Variational Problems.- 1.3. The Gradient Method for Optimization Problems with Differential Equations as Constraints.- 1.3.1. The Key Equation for Optimization Problems with Differential Equations as Constraints.- 1.3.2. Discussion of various Simple Problems.- 1.3.3. Constraints for Control Functions and State Variables.- 1.3.4. Treatment of the Maximum Climb of a Sounding Rocket by the Gradient Method.- 1.3.5. Discussion of the General Problem Formulation.- 1.3.6. Degree of Approximation of the Original Trajectories.- 2. Generalizations of the Gradient Method of the First Order and Related Methods.- 2.1. The Gradient Method of the 2nd Order.- 2.1.1. Definition of the Gradient Method of the 2nd Order.- 2.1.2. Derivation of the Formulas.- 2.2. Methods of Partial Expansion up to the 2nd Order.- 2.2.1. Basic Formulas of Partial Expansion.- 2.2.2. Solution of the General Problem in accordance with the Partial Expansion.- 2.2.3. The Extr. -H -Method.- 2.2.4. Generalizations of the Extr. -H -Method.- 2.3. Relationship between the Numerical Solution of the Necessary Conditions for an Optimum and the Gradient Method.- 2.3.1. Systematic Cross Connections.- 2.3.2. Comparison of the Quality of the Different Methods.- 3. The Bellman Dynamic Programming Method.- 3.1. The Bellman Method for Ordinary Extremal Problems.- 3.1.1. Preliminary Remarks.- 3.1.2. The Bellman Method in its Simplest Form.- 3.1.3. Solution of an Elementary Example.- 3.1.4. General Advantages of the Bellman Method and Comparison with the Systematic Search of a Grid.- 3.2. The Bellman Method for Simple Variational Problems.- 3.2.1. The Bellman Procedure.- 3.2.2. The Hamilton-Jacobi Differential Equation as Limiting Case of the Bellman Method.- 3.3. The Bellman Method for Optimization Problems with Differential Equations as Constraints.- 3.3.1. The Treatment of the Pontryagin Problem.- 3.3.2. Fundamental Discussion of an Example.- 3.3.3. Analytical Consideration of the Example Corresponding to a Passage to a Limit in the case of the Bellman Method, and according to the Pontryagin Maximum Principle.- 3.3.4. Conclusions.- 3.3.5. Variation Possibilities in the case of Bellman's Method.- 3.4. Numerical Aspects of the Bellman Method.- 3.4.1. Assessment of the Computation Effort.- 3.4.2. Polynomial Approximation to Reduce the Problem of Dimensions.- 3.4.3. Consideration of the Maximum Height of Climb of a Sounding Rocket by the Bellman Method.- 3.5. Linear Processes with Quadratic Performance Criteria.- 3.5.1. Basic Considerations.- 3.5.2. Practical Example.- Problems.- Comments.
