In the late 1950's, the group of Soviet mathematicians consisting of L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko made fundamental contributions to optimal control theory. Much of their work was collected in their monograph, The Mathematical Theory of Optimal Processes. Subsequently, Professor Gamkrelidze made further important contributions to the theory of necessary conditions for problems of optimal control and general optimization problems. In the present monograph, Professor Gamkrelidze presents his current view of the fundamentals of optimal control theory. It is intended for use in a one-semester graduate course or advanced undergraduate course. We are now making these ideas available in English to all those interested in optimal control theory. West Lafayette, Indiana, USA Leonard D. Berkovitz Translation Editor Vll Preface This book is based on lectures I gave at the Tbilisi State University during the fall of 1974. It contains, in essence, the principles of general control theory and proofs of the maximum principle and basic existence theorems of optimal control theory. Although the proofs of the basic theorems presented here are far from being the shortest, I think they are fully justified from the conceptual view point. In any case, the notions we introduce and the methods developed have one unquestionable advantage -they are constantly used throughout control theory, and not only for the proofs of the theorems presented in this book.

Inhalt

1. Formulation of the Time-Optimal Problem and Maximum Principle.- 1.1. Statement of the Optimal Problem.- 1.2. On the Canonical Systems of Equations Containing a Parameter and on the Pontryagin Maximum Condition.- 1.3. The Pontryagin Maximum Principle.- 1.4. A Geometrical Interpretation of the Maximum Condition..- 1.5. The Maximum Condition in the Autonomous Case.- 1.6. The Case of an Open Set U. The Canonical Formalism for the Solution of Optimal Control Problems.- 1.7. Concluding Remarks.- 2. Generalized Controls.- 2.1. Generalized Controls and a Convex Control Problem.- 2.2. Weak Convergence of Generalized Controls.- 3. The Approximation Lemma.- 3.1. Partition of Unity.- 3.2. The Approximation Lemma.- 4. The Existence and Continuous Dependence Theorem for Solutions of Differential Equations.- 4.1. Preparatory Material.- 4.2. A Fixed-Point Theorem for Contraction Mappings.- 4.3. The Existence and Continuous Dependence Theorem for Solutions of Equation (4.3).- 4.4. The Spaces ELip(G).- 4.5. The Existence and Continuous Dependence Theorems for Solutions of Differential Equations in the General Case.- 5. The Variation Formula for Solutions of Differential Equations.- 5.1. The Spaces Ex and Ex(G).- 5.2. The Equation of Variation and the Variation Formula for the Solution.- 5.3. Proof of Theorem 5.1.- 5.4. A Counterexample.- 5.5 On Solutions of Linear Matrix Differential Equations.- 6. The Varying of Trajectories in Convex Control Problems.- 6.1. Variations of Generalized Controls and the Corresponding Variations of the Controlled Equation.- 6.2. Variations of Trajectories.- 7. Proof of the Maximum Principle.- 7.1. The Integral Maximum Condition, the Pontryagin Maximum Condition, and Their Equivalence.- 7.2. The Maximum Principle in the Class of Generalized Controls.- 7.3. Construction of the Cone of Variations.- 7.4. Proof of the Maximum Principle.- 8. The Existence of Optimal Solutions.- 8.1. The Weak Compactness of the Class of Generalized Controls.- 8.2. The Existence Theorem for Convex Optimal Problems.- 8.3. The Existence Theorem in the Class of Ordinary Controls..- 8.4. Sliding Optimal Regimes.- 8.5. The Existence Theorem for Regular Problems of the Calculus of Variations.