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Summary
Summary
The control and estimation of continuous-time/continuous-space nonlinear systems continues to be a challenging problem, and this is one of the c- tral foci of this book. A common approach is to use dynamic programming; this typically leads to solution of the control or estimation problem via the solution of a corresponding Hamilton-Jacobi (HJ) partial di?erential eq- tion (PDE). This approach has the advantage of producing the "optimal" control. (The term "optimal" has a somewhat more complex meaning in the class of H problems. However, we will freely use the term for such controllers ? throughout, and this meaning will be made more precise when it is not ob- ous. )Thus,insolvingthecontrol/estimationproblem,wewillbesolvingsome nonlinear HJ PDEs. One might note that a second focus of the book is the solution of a class of HJ PDEs whose viscosity solutions have interpretations as value functions of associated control problems. Note that we will brie?y discuss the notion of viscosity solution of a nonlinear HJ PDE, and indicate that this solution has the property that it is the correct weak solution of the PDE. By correct weak solution in this context, we mean that it is the solution that is the value function of the associated control (or estimation) problem. The viscosity solution is also the correct weak solution in many PDE classes not considered here, and references to further literature on this subject will be given.
Table of Contents
Preface | p. vii |
1 Introduction | p. 1 |
1.1 Some Control and Estimation Problems | p. 3 |
1.2 Concepts of Max-Plus Methods | p. 6 |
2 Max-Plus Analysis | p. 11 |
2.1 Spaces of Semiconvex Functions | p. 13 |
2.2 Bases | p. 15 |
2.3 Two-Parameter Families | p. 21 |
2.4 Dual Spaces and Reflexivity | p. 22 |
3 Dynamic Programming and Viscosity Solutions | p. 31 |
3.1 Dynamic Programming Principle | p. 32 |
3.2 Viscosity Solutions | p. 42 |
4 Max-Plus Eigenvector Method for the Infinite Time-Horizon Problem | p. 57 |
4.1 Existence and Uniqueness | p. 58 |
4.2 Max-Plus Linearity of the Semigroup | p. 60 |
4.3 Semiconvexity and a Max-Plus Basis | p. 66 |
4.4 The Eigenvector Equation | p. 70 |
4.5 The Power Method | p. 72 |
4.6 Computing B: Initial Notes | p. 83 |
4.7 Outline of Algorithm | p. 84 |
4.8 A Control Problem Without Nominal Stability and a Game | p. 84 |
4.8.1 A Game Problem | p. 93 |
4.9 An Example | p. 95 |
5 Max-Plus Eigenvector Method Error Analysis | p. 97 |
5.1 Allowable Errors in Computation of B | p. 98 |
5.2 Convergence and Truncation Errors | p. 107 |
5.2.1 Convergence | p. 108 |
5.2.2 Truncation Error Estimate | p. 110 |
5.3 Errors in the Approximation of B | p. 119 |
5.3.1 A Method for Computing B | p. 122 |
5.4 Error Summary | p. 124 |
5.5 Example of Convergence Rate | p. 126 |
6 A Semigroup Construction Method | p. 129 |
6.1 Constituent Problems | p. 130 |
6.2 Operating on the Transformed Operators | p. 133 |
6.3 The HJB PDE Limit Problems | p. 134 |
6.4 A Simple Example | p. 138 |
7 Curse-of-Dimensionality-Free Method | p. 143 |
7.1 DP for the Constituent and Originating Problems | p. 146 |
7.2 Max-Plus Spaces and Dual Operators | p. 150 |
7.3 Discrete Time Approximation | p. 158 |
7.4 The Algorithm | p. 164 |
7.5 Practical Issues | p. 170 |
7.5.1 Pruning | p. 170 |
7.5.2 Initialization | p. 171 |
7.6 Examples | p. 171 |
7.7 More General Quadratic Constituents | p. 175 |
7.8 Future Directions | p. 180 |
8 Finite Time-Horizon Application: Nonlinear Filtering | p. 183 |
8.1 Semiconvexity | p. 187 |
8.2 Max-Plus Propagation | p. 192 |
9 Mixed L[subscript infinity]/L[subscript 2] Criteria | p. 197 |
9.1 Mixed L[subscript infinity]/L[subscript 2] Problem Formulation | p. 197 |
9.2 Dynamic Programming | p. 200 |
9.2.1 Dynamic Programming Principles | p. 200 |
9.2.2 Dynamic Programming Equations | p. 204 |
9.3 Max-Plus Representations and Semiconvexity | p. 205 |
9.4 Max-Plus Numerical Methods | p. 209 |
9.4.1 Nonuniqueness for the Max-Plus Affine Equation | p. 211 |
9.4.2 The Affine Power Method | p. 212 |
A Miscellaneous Proofs | p. 217 |
A.0.1 Sketch of Proof of Theorem 2.8 | p. 217 |
A.0.2 Proof of Theorem 3.13 | p. 218 |
A.0.3 Proof of Lemma 3.15 | p. 220 |
A.0.4 Sketch of Proof of Theorem 7.27 | p. 222 |
A.0.5 Sketch of Proof of Lemma 7.31 | p. 224 |
A.0.6 Existence of Robust/H[subscript infinity] Estimator and a Disturbance Bound | p. 228 |
References | p. 233 |
Index | p. 239 |