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. Language: English