Tabla de Contenidos: “….; 2.8 Problemas y ejercicios; Capítulo 3 Funciones elementales; 3.1 Funciones elementales básicas; 3.1.1 Introducción Leonhard Euler (1707-1783)…”

Tabla de Contenidos: “…3.3 CURVAS CIRCULARES COMPUESTAS3.3.1 Curvas circulares compuestas de dos radios; 3.3.2 Curvas circulares compuestas de tres radios; 3.4 ESTABILIDAD EN LA MARCHA, VELOCIDAD, CURVATURA, PERALTE Y TRANSICIÓN; 3.4.1 Velocidad de diseño; 3.4.2 Velocidad específica; 3.4.3 Desplazamiento de un vehículo sobre una curva circular; 3.4.4 Velocidad, curvatura, peralte y fricción lateral; 3.4.5 Transición del peralte; 3.5 CURVAS ESPIRALES DE TRANSICIÓN; 3.5.1 Generalidades; 3.5.2 La espiral de Euler o Clotoide como curva de transición; 3.5.3 Ecuaciones de la Clotoide o espiral de transición…”

Tabla de Contenidos: “…4.1.2 Connectedness with respect to a subgraph4.1.3 Adjacency graphs; 4.1.4 Types of nodes; region adjacencies; 4.2 Some Basics of Graph Theory; 4.2.1 Nodes, paths, and distances; 4.2.2 Special types of nodes, edges, and graphs; 4.3 Oriented Adjacency Graphs; 4.3.1 Local circular orders; 4.3.2 The Euler characteristic and planarity; 4.3.3 Atomic and border cycles; 4.3.4 The separation theorem; 4.3.5 Holes; 4.3.6 Boundaries; 4.3.7 Some combinatorial results; 4.4 Combinatorial Maps; 4.4.1 2D maps; 4.4.2 3D maps; 4.5 Exercises; 4.6 Commented Bibliography; 5. …”

Tabla de Contenidos: “…Exercise 3.7 -- Exercise 3.8 -- Exercise 3.9 -- 3.2.2 Linear Equations with Variable Coefficients- Euler-Cauchy Equations (Equidimensional Equations) -- Exercise 3.10 -- 3.2.3 Legendre's Linear Equation -- Exercise 3.11 -- 3.2.4 Method of Variation of Parameters -- Exercise 3.12 -- 3.2.5 Systems of Simultaneous Linear Differential Equations with Constant Coefficients -- Exercise 3.13 -- Chapter 4: Differential Equations of the First Order but not of the First Degree -- 4.1 Equations solvable for p -- Exercise 4.1 -- 4.2 Equations solvable for y -- Exercise 4.2 -- 4.3 Equations solvable for x -- Exercise 4.3 -- Chapter 5: Linear Equation of the Second Order with Variable Coefficients -- 5.1 To find the integral in C.F. by inspection, i.e. to find a solution of -- Exercise 5.1 -- 5.2 General solution of R by changing the dependent variable and removing the first derivative (Reduction to normal form) -- Exercise 5.2 -- 5.3 General solution of by changing -- Exercise 5.3 -- Chapter 6: Integration in Series: Legendre, Bessel and Chebyshev Functions -- 6.1 Legendre functions -- 6.1.1 Introduction -- 6.1.2 Power Series Method of Solution of Linear Differential Equations -- 6.1.3 Existence of Series Solutions: Method of Frobenius -- 6.1.4 Legendre Functions -- 6.1.5 Legendre Polynomials Pa(x) -- 6.1.6 Generating Function for Legendre Polynomials Pn(x) -- 6.1.7 Recurrence Relations of Legendre Functions -- 6.1.8 Orthogonality of Functions -- 6.1.9 Orthogonality of Legendre Polynomials Pn(x) -- 6.1.10 Betrami's Result -- 6.1.11 Christoffel's Expansion -- 6.1.12 Christoffel's Summation Formula -- 6.1.13 Laplace's First Integral for (x) -- 6.1.14 Laplace's Second Integral for (x) -- 6.1.15 Expansion of f(x) in a Series of Legendre Polynomials -- Exercise 6.1 -- 6.2 Bessel functions -- 6.2.1 Introduction -- 6.2.2 Bessel Functions…”

Tabla de Contenidos: “…ASPECTOS ESTRUCTURALES ADICIONALES; 4.2.3. DECONVOLUCIÓN DE EULER; 4.3. IMÁGENES DE POTASIO, TORIO Y URANIO; 4.3.1. …”

Tabla de Contenidos: “…Numerical Methods, Calculus, and Statistics -- 6.1 Introduction -- 6.2 Numerical Integration -- 6.2.1 Double and Triple Integrals -- 6.3 Numerical Differentiation -- 6.4 Interpolation -- 6.4.1 Multidimensional Interpolation -- 6.5 Curve Fitting -- 6.6 Numerical Optimization -- 6.7 Linear Equations -- 6.7.1 Gauss Elimination -- 6.7.2 Ill-conditioned Problems -- 6.7.3 Pivoting to Handle Ill-conditioned Problems -- 6.8 Solution of Nonlinear Algebraic Equations -- 6.8.1 Gauss-Seidel (GS) Method -- 6.8.2 GS Solution for a System of Equations -- 6.8.3 Newton-Raphson (NR) Method -- 6.8.4 NR Solution for a System of Equations -- 6.9 Numerical Solution of Ordinary Differential Equations -- 6.9.1 Euler's Method for Solution of Differential Equations -- 6.9.2 Euler's Modified Method…”

Tabla de Contenidos: “….) -- CONDICIONES INICIALES -- MÉTODO DE EULER (RUNGE-KUTTA DE PRIMER (...) -- MÉTODO RUNGE-KUTTA DE SEGUNDO ORDEN -- MÉTODO RUNGE-KUTTA DE CUARTO ORDEN -- MÉTODO DE TAYLOR DE SEGUNDO ORDEN -- MÉTODO DE TAYLOR DE CUARTO ORDEN -- EJEMPLO IQUI -- CONDICIONES DE FRONTERA -- SISTEMAS DE UNA CONDICIÓN DE FRONTERA -- SISTEMAS DE DOS O MAS CONDICIONES DE FRONTERA -- EJEMPLO IQUI -- SISTEMAS DE ECUACIONES ALGEBRO-DIFERENCIALES (...) -- SOLUCIÓN DE SISTEMAS DAE DE ÍNDICE 1 -- SOLUCIÓN DE SISTEMAS DAE DE ÍNDICE 2 O (...) -- SISTEMAS DE ECUACIONES DIFERENCIALES PARCIALES (...) -- MÉTODO DE DIFERENCIAS FINITAS -- EJEMPLO IQUI -- MESH -- ALGORITMO DE THOMAS -- REFERENCIAS -- LOS AUTORES…”

Tabla de Contenidos: “…-- Hartmut Wedekind Es geht nicht darum, schnell in die abstrakte Welt der Begriffe, Formeln und Modelle einzutauchenAnhang zum Buch I "Die Vergangenheit verstehen" -- I Biographische Daten Martin Wagenscheins -- II Publikationen von Martin Wagenschein sowie Beiträge zur Pädagogik und Didaktik Wagenscheins -- III Das Wagenscheinarchiv.Klaus Kohl Das Wagenscheinarchiv -- IV Die Schweizerische Wagenschein-Gesellschaft -- Peter Stettler Erinnerungssplitter an die Schweizerische Wagenschein-Gesellschaft -- V Die Wagenscheintagungen und die Wagenschein-Preise -- Peter Stettler Die internationalen Wagenscheintagungen 1987-2000 -- Ueli Aeschlimann Wagenscheinkurse und Wagenschein-Tagungen in LiestalPeter Buck, Marc Müller & Svantje Schumann Rückblick und Ausblick -- VI Ringvorlesungen Zweites Buch: "Die Zukunft gestalten" Marc Müller & Svantje Schumann Einleitung Peter Labudde Martin Wagenschein ein Wegbereiter des KonstruktivismusLutz-Helmut Schön Brücken bauen mit Martin Wagenschein -- Uwe Hericks Genetisch-sokratisch Lehren eine Didaktik zur Professionalisierung von Lehrerinnen und LehrernPeter Euler Verstehen als pädagogische Kategorie Am Beispiel subjektiver Sach- und Facherschließung der Naturwissenschaften -- Svantje Schumann Professionalisierung, pädagogisches Arbeitsbündnis und lebendige Bildungsprozesserfahrung bei Wagenschein -- Ueli Aeschlimann & Nicola Meschede Wagenscheins sokratisches Gespräch Erfahrungen aus der Umsetzung im Unterricht und in der Lehrerausbildung -- Peter Stettler Wissenschaft sverständigkeit Martin Wagenscheins wissenschaft stheoretische Impulse -- Franz Arndt Wagenschein und die Waldorfpädagogik Versuch einer Begegnung -- Martin Gröger & Katharina Wurm Mit Wagenschein auf dem Weg zu den kleinsten TeilchenDieter Plappert Wagenschein aktuell die Freiburger Forschungsräume -- Alexandria Krug, Ruedi Küng & Dieter Franz Obermaier Schulgärten -- 3: exemplarisch genetisch sokratisch In Schulgärten mit den Ansätzen Martin Wagenscheins unterwegs -- Florian Theilmann Zum Problem des Genetischen Unterrichtens in den Naturwissenschaften -- Hubert Schnüriger Sokratische Gespräche und fachliche Erkenntnisziele Ein Vergleich -- Andreas Schulz & Stefan Brackertz Martin Wagenschein in der Lehramtsausbildung -- Autorinnen und Autoren…”

Tabla de Contenidos: “…13.8 Strut Fixed at Both Ends (Ends Direction Fixed) -- 13.9 Euler Theory and Experimental Results -- 13.10 Rankine-Gordon Formula -- 13.11 Other Formulae -- 13.12 A Strut with Eccentric Load -- 13.13 Laterally Loaded Struts -- Problems -- Objective Questions -- Answers -- Chapter 14: Thick Cylinders and Spheres -- 14.1 Stresses in Thick Cylinders -- 14.2 Lame's Equations -- 14.3 Special Case of Solid Shaft -- 14.4 Strains and Displacements, Compound Cylinders -- 14.5 Thick Spherical Shells -- Problems -- Objective Questions -- Answers -- Chapter 15: Rotating Rings, Discs and Cylinders -- 15.1 Rotating Ring -- 15.2 Rotating Disc -- 15.3 Special Cases of Rotating Disc -- 15.4 Disc of Uniform Strength -- 15.5 Rotating Cylinder -- 15.6 Axial Stress in Rotating Cylinder-Hollow Cylinder -- 15.7 Axial Stress in Solid Rotating Cylinder -- Problems -- Objective Questions -- Answers -- Chapter 16: Curved Beams -- 16.1 Introduction -- 16.2 Stresses in Curved Beams -- 16.3 Determination of Eccentricity -- 16.4 Rings Under Load -- 16.4.1 Deflection of Rings in the Direction of Load -- 16.4.2 Deflection of Ring Under Transverse Load -- 16.5 Stresses in Rings -- 16.6 Chain Link -- Problems -- Objective Questions -- Answer -- Chapter 17: Vibrations -- 17.1 Vibrations Types -- 17.2 Simple Harmonic Motion -- 17.3 Linear Vibration -- 17.4 Transverse Vibration -- 17.4.1 Inertia of the Bar -- 17.5 Transverse Vibration of a Shaft Under its Own Weight -- 17.5.1 Several Loads on a Beam -- 17.6 Torsional Vibration -- 17.7 Torsional Vibration of Shaft Under Two Rotors -- 17.8 Helical Spring Under Torsion -- 17.9 Whirling of Shaft -- 17.9.1 Shaft Carrying a Single Rotor with Eccentricity -- Problems -- Objective Questions -- Answers -- Appendix: Theory of Elasticity -- A-1 Differential Equation of Equilibrium -- A.2. …”

Tabla de Contenidos: “…2.4.10 Application of Fractional Calculus to Fluid Mechanics -- 2.4.11 Radioactivity, Exponential Decay, and Population Growth -- 2.4.12 The Harmonic Oscillator -- 2.5 Overview of Some Analytical/Numerical Methods -- 2.5.1 Fractional Adams-Bashforth/Moulton Methods -- 2.5.2 Fractional Euler Method -- 2.5.3 Finite Difference Method -- 2.5.4 Finite Element Method -- 2.5.5 Finite Volume Method -- 2.5.6 Meshless Method -- 2.5.7 Reproducing Kernel Hilbert Space Method -- 2.5.8 Wavelet Method -- 2.5.9 The Sine-Gordon Expansion Method -- 2.5.10 The Jacobi Elliptic Equation Method -- 2.5.11 The Generalized Kudryashov Method -- References -- Chapter 3 Adomian Decomposition Method -- 3.1 Introduction -- 3.2 Basic Idea of ADM -- 3.3 Numerical Examples -- References -- Chapter 4 Adomian Decomposition Transform Method -- 4.1 Introduction -- 4.2 Transform Methods for the Caputo Sense Derivatives -- 4.3 Adomian Decomposition Laplace Transform Method (ADLTM) -- 4.4 Adomian Decomposition Sumudu Transform Method (ADSTM) -- 4.5 Adomian Decomposition Elzaki Transform Method (ADETM) -- 4.6 Adomian Decomposition Aboodh Transform Method (ADATM) -- 4.7 Numerical Examples -- 4.7.1 Implementation of ADLTM -- 4.7.2 Implementation of ADSTM -- 4.7.3 Implementation of ADETM -- 4.7.4 Implementation of ADATM -- References -- Chapter 5 Homotopy Perturbation Method -- 5.1 Introduction -- 5.2 Procedure for HPM -- 5.3 Numerical Examples -- References -- Chapter 6 Homotopy Perturbation Transform Method -- 6.1 Introduction -- 6.2 Transform Methods for the Caputo Sense Derivatives -- 6.3 Homotopy Perturbation Laplace Transform Method (HPLTM) -- 6.4 Homotopy Perturbation Sumudu Transform Method (HPSTM) -- 6.5 Homotopy Perturbation Elzaki Transform Method (HPETM) -- 6.6 Homotopy Perturbation Aboodh Transform Method (HPATM) -- 6.7 Numerical Examples -- 6.7.1 Implementation of HPLTM…”

Tabla de Contenidos: “…Ianvae et in Monte Pessulano anno 1303 composita/ edidit Walter Euler. -- T. XXIV: 65. Arbor Scientiae. Romae in festo sancti Michaelis Archangeli, anno MCCXCV incepta in ipsa urbe Kalendis aprilibus: anni MCCXCVI ad finem perducta. …”

Tabla de Contenidos: “…DIFFERENTIAL EQUATIONS. Slope Fields and Euler's Method. Differential Equations: Growth and Decay. …”

Tabla de Contenidos: “…List of Figures xiii -- List of Tables xix -- Preface xxi -- Acknowledgments xxv -- Part I Neural Networks for Serial Robot Arm Control 1 -- 1 Zeroing Neural Networks for Control 3 -- 1.1 Introduction 3 -- 1.2 Scheme Formulation and ZNN Solutions 4 -- 1.2.1 ZNN Model 4 -- 1.2.2 Nonconvex Function Activated ZNN Model 8 -- 1.3 Theoretical Analyses 9 -- 1.4 Computer Simulations and Verifications 12 -- 1.4.1 ZNN for Solving (1.13) at t = 1 12 -- 1.4.2 ZNN for Solving (1.13) with Different Bounds 15 -- 1.5 Summary 16 -- 2 Adaptive Dynamic Programming Neural Networks for Control 17 -- 2.1 Introduction 17 -- 2.2 Preliminaries on Variable Structure Control of the Sensor-Actuator System 18 -- 2.3 Problem Formulation 19 -- 2.4 Model-Free Control of the Euler-Lagrange System 20 -- 2.4.1 Optimality Condition 21 -- 2.4.2 Approximating the Action Mapping and the Critic Mapping 21 -- 2.5 Simulation Experiment 23 -- 2.5.1 The Model 23 -- 2.5.2 Experiment Setup and Results 24 -- 2.6 Summary 25 -- 3 Projection Neural Networks for Robot Arm Control 27 -- 3.1 Introduction 27 -- 3.2 Problem Formulation 29 -- 3.3 A Modified Controller without Error Accumulation 30 -- 3.3.1 Existing RNN Solutions 30 -- 3.3.2 Limitations of Existing RNN Solutions 32 -- 3.3.3 The Presented Algorithm 33 -- 3.3.4 Stability 34 -- 3.4 Performance Improvement Using Velocity Compensation 36 -- 3.4.1 A Control Law with Velocity Compensation 36 -- 3.4.2 Stability 37 -- 3.5 Simulations 41 -- 3.5.1 Regulation to a Fixed Position 41 -- 3.5.2 Tracking of Time-Varying References 42 -- 3.5.3 Comparisons 47 -- 3.6 Summary 50 -- 4 Neural Learning and Control Co-Design for Robot Arm Control 51 -- 4.1 Introduction 51 -- 4.2 Problem Formulation 52 -- 4.3 Nominal Neural Controller Design 53 -- 4.4 A Novel Dual Neural Network Model 54 -- 4.4.1 Neural Network Design 54 -- 4.4.2 Stability 56 -- 4.5 Simulations 62 -- 4.5.1 Simulation Setup 62 -- 4.5.2 Simulation Results 63 -- 4.5.2.1 Tracking Performance 63 -- 4.5.2.2 With vs.Without Excitation Noises 64.…”

Tabla de Contenidos: “…Front Cover -- Handbook of Computational Economics -- Copyright -- Contents -- Contributors -- Introduction to the Series -- Introduction to the Handbook of Computational Economics, Volume 4, Heterogeneous Agent Modeling -- 1 Introduction -- 2 Macroeconomics -- 3 Finance -- 4 Experiments -- 5 Networks -- 6 Other -- Acknowledgments -- References -- Part 1 Macroeconomics -- 1 Heterogeneous Expectations and Micro-Foundations in Macroeconomics -- 1 Introduction -- 2 Expectations Operators and Bounded Rationality -- 2.1 Expectations Operators -- 2.2 The Economic Environment -- 2.3 Bounded Optimality -- 2.3.1 Rational Expectations -- 2.3.2 The Shadow-Price Approach -- 2.3.3 The Shadow-Price Approach in the Linearized Model -- 2.3.4 The Euler Equation Approach in the Linearized Model -- 2.3.5 The Finite Horizon Approach in the Linearized Model -- 2.3.6 The In nite Horizon Approach in the Linearized Model -- 2.3.7 A Nod to Value Functions -- 2.3.8 A Defense of Anticipated Utility -- 2.4 Aggregating Household Decision Rules -- 3 Equilibria with Heterogeneous Expectations -- 3.1 Extrinsic Heterogeneity -- 3.2 Rationally Heterogeneous Expectations -- 3.2.1 Predictor Selection -- 3.2.2 Heterogeneous Beliefs and Economic Dynamics: Stability Reversal -- 3.3 Intrinsic Heterogeneity -- 4 Asset-Pricing Applications -- 4.1 Regime-Switching Returns -- 4.2 Bubbles with Rationally Heterogeneous Expectations -- 4.3 Restricted Perceptions and Endogenous Fluctuations -- 4.4 Related Literature -- 5 Monetary Applications -- 5.1 A Monetary Search Model with Heterogeneous Expectations -- 5.2 Heterogeneous Beliefs and Bargaining -- 5.3 Equilibrium with Heterogeneous Beliefs -- 5.4 Uncertainty and Welfare -- 5.5 Related Literature -- 6 DSGE Applications -- 6.1 Rationally Heterogeneous Expectations and Monetary Policy Rules…”

Tabla de Contenidos: “…Front Cover -- Table of Integrals, Series, and Products -- Copyright -- Contents -- Preface to the Eighth Edition -- Acknowledgments -- The Order of Presentation of the Formulas -- Use of the Tables -- Bernoulli and Euler Polynomials and Numbers -- Elliptic Functions and Elliptic Integrals -- The Jacobi Zeta Function and Theta Functions -- Exponential and Related Integrals -- Hermite and Chebyshev Orthogonal Polynomials -- Bessel Functions -- Parabolic Cylinder Functions and Whittaker Functions -- Mathieu Functions -- Index of Special Functions -- Notation -- Note on the Bibliographic References -- 0 Introduction -- 0.1 Finite sums -- 0.11 Progressions -- 0.12 Sums of powers of natural numbers -- 0.13 Sums of reciprocals of natural numbers -- 0.14 Sums of products of reciprocals of natural numbers -- 0.15 Sums of the binomial coefficients -- 0.2 Numerical series and infinite products -- 0.21 The convergence of numerical series -- 0.22 Convergence tests -- 0.23-0.24 Examples of numerical series -- 0.25 Infinite products -- 0.26 Examples of infinite products -- 0.3 Functional series -- 0.30 Definitions and theorems -- 0.31 Power series -- 0.32 Fourier series -- 0.33 Asymptotic series -- 0.4 Certain formulas from differential calculus -- 0.41 Differentiation of a definite integral with respect to a parameter -- 0.42 The nth derivative of a product (Leibniz's rule) -- 0.43 The nth derivative of a composite function -- 0.44 Integration by substitution -- 1 Elementary Functions -- 1.1 Power of Binomials -- 1.11 Power series -- 1.12 Series of rational fractions -- 1.2 The Exponential Function -- 1.21 Series representation -- 1.22 Functional relations -- 1.23 Series of exponentials -- 1.3-1.4 Trigonometric and Hyperbolic Functions -- 1.30 Introduction -- 1.31 The basic functional relations…”

Tabla de Contenidos: “…Incompressible flow estimates of acoustic source terms -- Chapter 6: The linearized Euler equations -- 6.1. Goldstein's equation -- 6.2. …”

Tabla de Contenidos:

Tabla de Contenidos: “…Intro -- Title Page -- Copyright Page -- Contents -- Preface -- Dimitriadis: Nonlinear Aeroelasticity - Series Preface Oct 2016 -- About the Companion Website -- Chapter 1 Introduction -- 1.1 Sources of Nonlinearity -- 1.2 Origins of Nonlinear Aeroelasticity -- References -- Chapter 2 Nonlinear Dynamics -- 2.1 Introduction -- 2.2 Ordinary Differential Equations -- 2.3 Linear Systems -- 2.3.1 Stable Oscillatory Response -- 2.3.2 Neutral Oscillatory Response -- 2.3.3 Unstable Oscillatory Response -- 2.3.4 Stable Non-oscillatory Response -- 2.3.5 Unstable Non-oscillatory Response -- 2.3.6 Fixed Point Summary -- 2.4 Nonlinear Systems -- 2.4.1 Linearisation Around Fixed Points -- 2.4.2 The Pitching Wing Section with Cubic Stiffness -- 2.4.3 The Pitchfork Bifurcation -- 2.5 Stability in the Lyapunov Sense -- 2.6 Asymmetric Systems -- 2.6.1 The Fold Bifurcation -- 2.6.2 The Transcritical Bifurcation -- 2.7 Existence of Periodic Solutions -- 2.7.1 Nonlinear Aeroelastic Galloping -- 2.8 Estimating Periodic Solutions -- 2.8.1 Periodic Solutions of the Nonlinear Galloping Oscillator -- 2.8.2 The Hopf Bifurcation -- 2.9 Stability of Periodic Solutions -- 2.9.1 Stability of Galloping Oscillations -- 2.9.3 The Fold Bifurcation of Cycles -- 2.10 Concluding Remarks -- References -- Chapter 3 Time Integration -- 3.1 Introduction -- 3.2 Euler Method -- 3.2.1 Linear Systems -- 3.2.2 Nonlinear Systems -- 3.3 Central Difference Method -- 3.3.1 Explicit Solution of Nonlinear Systems -- 3.3.2 Implicit Solution of Nonlinear Systems -- 3.4 Runge-Kutta Method -- 3.5 Time-Varying Linear Approximation -- 3.6 Integrating Backwards in Time -- 3.7 Time Integration of Systems with Multiple Degrees of Freedom -- 3.8 Forced Response -- 3.9 Harmonic Balance -- 3.9.1 Newton-Raphson -- 3.9.2 Discrete Fourier Transform Techniques -- 3.10 Concluding Remarks -- References…”

Tabla de Contenidos: “…EJERCICIOS PROPUESTOS -- CAPÍTULO 22. ECUACIONES DE CAUCHY-EULER -- 22.1. SI LA E.D. ES HOMOGÉNEA -- 22.1.1. …”

Tabla de Contenidos: “…6.8.4.3 Adam -- 6.9 Coordinate Descent Methods -- 6.10 Interior‐Point Methods -- 6.10.1 Path‐Following Method -- 6.10.2 Generalized Inequalities -- 6.11 Augmented Lagrangian Methods -- 6.11.1 Method of Multipliers -- 6.11.2 Alternating Direction Method of Multipliers -- 6.11.3 Variable Splitting -- Exercises -- Part III Optimal Control -- Chapter 7 Calculus of Variations -- 7.1 Extremum of Functionals -- 7.1.1 Necessary Condition for Extremum -- 7.1.2 Sufficient Condition for Optimality -- 7.1.3 Constrained Problem -- 7.1.4 Du Bois-Reymond Lemma -- 7.1.5 Generalizations -- 7.2 The Pontryagin Maximum Principle -- 7.2.1 Linear Quadratic Control -- 7.2.2 The Riccati Equation -- 7.3 The Euler-Lagrange Equations -- 7.3.1 Beltrami's Identity -- 7.4 Extensions -- 7.5 Numerical Solutions -- 7.5.1 The Gradient Method -- 7.5.2 The Shooting Method -- 7.5.3 The Discretization Method -- 7.5.4 The Multiple Shooting Method -- 7.5.5 The Collocation Method -- Exercises -- Chapter 8 Dynamic Programming -- 8.1 Finite Horizon Optimal Control -- 8.1.1 Standard Optimization Problem -- 8.1.2 Dynamic Programming -- 8.2 Parametric Approximations -- 8.2.1 Fitted‐Value Iteration -- 8.3 Infinite Horizon Optimal Control -- 8.3.1 Bellman Equation -- 8.4 Value Iterations -- 8.5 Policy Iterations -- 8.5.1 Approximation -- 8.6 Linear Programming Formulation -- 8.6.1 Approximations -- 8.7 Model Predictive Control -- 8.7.1 Infinite Horizon Problem -- 8.7.2 Guessing the Value Function -- 8.7.3 Finite Horizon Approximation -- 8.7.4 Receding Horizon Approximation -- 8.8 Explicit MPC -- 8.9 Markov Decision Processes -- 8.9.1 Stochastic Dynamic Programming -- 8.9.2 Infinite Time Horizon -- 8.9.3 Stochastic Bellman Equation -- 8.10 Appendix -- 8.10.1 Stability and Optimality of Infinite Horizon Problem -- 8.10.2 Stability and Optimality of Stochastic Infinite Time Horizon Problem…”