Author | Gamkrelidze, R. V. author |
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Title | Principles of Optimal Control Theory [electronic resource] / by R. V. Gamkrelidze |
Imprint | Boston, MA : Springer US : Imprint: Springer, 1978 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-7398-8 |
Descript | XII, 175 p. online resource |
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