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.