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An innovative and unique application of interval analysis to optimal control problems In Interval Analysis: Application in the Optimal Control Problems, celebrated researcher and engineer Dr. Navid Razmjooy delivers an expert discussion of the uncertainties in the analysis of optimal control problems. In the book, Dr. Razmjooy uses an open-ended approach to solving optimal control problems with indefinite intervals. Utilizing an extended, Runge-Kutta method, the author demonstrates how to accelerate its speed with the piecewise function. You'll find recursive methods used to achieve more…mehr
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An innovative and unique application of interval analysis to optimal control problems In Interval Analysis: Application in the Optimal Control Problems, celebrated researcher and engineer Dr. Navid Razmjooy delivers an expert discussion of the uncertainties in the analysis of optimal control problems. In the book, Dr. Razmjooy uses an open-ended approach to solving optimal control problems with indefinite intervals. Utilizing an extended, Runge-Kutta method, the author demonstrates how to accelerate its speed with the piecewise function. You'll find recursive methods used to achieve more compact answers, as well as how to solve optimal control problems using the interval Chebyshev's function. The book also contains: * A thorough introduction to common errors and mistakes, generating uncertainties in physical models * Comprehensive explorations of the literature on the subject, including Hukurara's derivatives * Practical discussions of the interval analysis and its variants, including the classical (Minkowski) methods * Complete treatments of existing control methods, including classic, conventional advanced, and robust control. Perfect for master's and PhD students working on system uncertainties, Interval Analysis: Application in the Optimal Control Problems will also benefit researchers working in laboratories, universities, and research centers.
Produktdetails
- Produktdetails
- Verlag: Wiley
- Seitenzahl: 208
- Erscheinungstermin: 27. Dezember 2023
- Englisch
- Abmessung: 235mm x 157mm x 16mm
- Gewicht: 464g
- ISBN-13: 9781394190973
- ISBN-10: 1394190972
- Artikelnr.: 68802273
- Verlag: Wiley
- Seitenzahl: 208
- Erscheinungstermin: 27. Dezember 2023
- Englisch
- Abmessung: 235mm x 157mm x 16mm
- Gewicht: 464g
- ISBN-13: 9781394190973
- ISBN-10: 1394190972
- Artikelnr.: 68802273
Navid Razmjooy, PhD, is an independent researcher based in Belgium. He holds a Ph.D. in Electrical Engineering (Control and Automation) from Tafresh University in Iran. His research is focused on renewable energies, interval analysis, image processing, machine vision, data mining, evolutionary algorithms, and system control.
About the Author viii 1 Preface and Overview 1 Chapter 1: Preface and
Overview 1 Chapter 2: Introduction 2 Chapter 3: Literature Review 2 Chapter
4: Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 2 Chapter 5: Stability and Controllability Based on Interval
Analysis 3 Chapter 6: Optimal Control of the Systems with Interval
Uncertainties 3 Chapter 7: Conclusions 3 2 Introduction 5 2.1 Background 5
2.2 Relationship Between Error and Uncertainty 7 2.3 Expert Perspectives on
Interval Analysis 8 2.4 Precision Analysis of Interval-Based Models
(Self-Validated Numeric) 10 2.5 Concepts of the Ordinary and Interval-based
Optimal Control 11 2.6 Orthogonal Spectral Methods 13 2.7 Desired
Confidence Interval 13 2.8 Conclusions 14 3 Literature Review 17 4
Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 29 4.1 Introduction 29 4.2 Introduction to IA 30 4.3 The
Algebra of Interval Sets 31 4.3.1 Classic Interval Algebra (Minkowski
Method) 31 4.3.2 Mathematical Norm and Distance in the IA 33 4.3.3 Kaucher
Extended Interval Analysis 33 4.3.4 Modal Interval Analysis 34 4.3.5
Hukuhara Difference Method 35 4.4 Interval Representations 36 4.5 Interval
Complex Integers 39 4.6 Interval Vector and Matrix 41 4.6.1 Vector and
Matrix IA 41 4.6.2 Interval Vector Norm 42 4.6.3 Interval Matrix Analysis
42 4.6.3.1 Conceptional Example: Wrapping Effect 43 4.6.4 Interval Matrices
Norm 44 4.6.5 Interval Determination of a Matrix 45 4.6.6 The Inverse of a
Regular Interval Matrix 45 4.6.7 Eigenvalues and Eigenvectors of an
Interval Matrix 45 4.7 Solving Linear Systems with Interval Parameters 46
4.8 Interval Functions 46 4.8.1 Overestimated Interval Function 46 4.8.2
Minimal Interval Function 48 4.9 Determining the Minimal Interval 49 4.9.1
Uniform Interval Functions 49 4.9.2 Nonuniform Interval Functions 49 4.9.3
Interval Power Series 50 4.10 Interval Derivative and Integral Functions 52
4.10.1 Interval Derivative 52 4.10.2 Interval Integration 53 4.11 Centered
Inclusion Method 54 4.11.1 Linearized Interval Functions Around the Center
54 4.11.2 Taylor Inclusion Functions 54 4.12 Interval Nonlinear Systems 56
4.13 Analysis of the Interval Dynamic Systems in the Presence of Interval
Uncertainties 57 4.13.1 Solving the Interval Initial Value Problems 58 4.14
The Interval Runge-Kutta Method (IRKM) for Interval Differential Equations
60 4.14.1 Introduction 60 4.14.2 Generalized IRKM (GIRKM) Based on
Switching Points 63 4.14.3 Numerical Examples 66 4.15 Interval Uncertainty
Analyses based on Orthogonal Functions 78 4.15.1 Interval
Epsilon-orthogonal 79 4.15.2 Interval Weierstrass's Theorem 79 4.16
Interval Orthogonal Polynomials 79 4.16.1 Legendre Polynomials 80 4.16.2
Chebyshev Polynomials 80 4.16.3 Interval Orthogonal Functions 82 4.17
Piecewise Extension of the Interval Orthogonal Functions 83 4.18 Conclusion
85 5 Stability and Controllability Based on Interval Analysis 91 5.1
Introduction 91 5.1.1 Classical Control Theory 91 5.1.2 Advanced Modern
Control Systems Theory 92 5.1.3 Optimal Control 93 5.1.4 Robust Control 94
5.1.5 Adaptive Control Theory 95 5.2 Interval Stability and Controllability
95 5.3 Interval Stability 96 5.4 Characteristic Polynomial 97 5.5
Routh-Hurwitz Stability Test 98 5.6 Kharitonov's Theorem (Interval
Routh-Hurwitz Stability Test) 99 5.6.1 Kharitonov Polynomial Theory 99
5.6.2 A Centered Representation of the Interval Routh-Hurwitz Stability
Criterion 102 5.7 Interval Stability Based on Linear Matrix Inequalities
102 5.7.1 The Positive Matrix of the Interval Matrix 102 5.7.2 Stability
Analysis of the Interval Systems 103 5.7.3 Linear Matrix Inequalities 104
5.8 Controllability and Observability 107 5.9 Controllability and
Observability Based on Interval Criteria 107 5.9.1 Singular Values for
Analyzing the Controllability 111 5.10 Conclusions 117 6 Optimal Control of
the Systems with Interval Uncertainties 121 6.1 Introduction 121 6.2
Indirect Methods 123 6.3 Direct Methods 123 6.4 Optimal Control Problem in
the Presence of Interval Uncertainties 127 6.5 Interval Optimal Control
Based on the Indirect Method 128 6.5.1 Analysis of the Standard Interval
Calculus of Variations 129 6.6 Analysis of the Problem of the Interval
Optimal Control Based on Euler-Lagrange Equations 131 6.7 Solving Optimal
Control Problems with Interval Uncertainties: Interval Runge-Kutta Method
132 6.8 Optimal Control of Problems with Interval Uncertainties Using the
Chebyshev Inclusion Method 146 6.9 Piecewise Interval Chebyshev Method for
OCPs 152 6.10 Solving Quadratic Optimal Control Problems with Interval
Uncertainties Based on Indirect Method: Interval Quadratic Regulator 159
6.11 Problem Statement (Interval Quadratic Regulator) 173 6.12 Interval
Optimal Control Based on Direct Method 179 6.13 Applied Simulations 183
6.14 Conclusion 192 References 193 7 Conclusions 197 Index 199
Overview 1 Chapter 2: Introduction 2 Chapter 3: Literature Review 2 Chapter
4: Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 2 Chapter 5: Stability and Controllability Based on Interval
Analysis 3 Chapter 6: Optimal Control of the Systems with Interval
Uncertainties 3 Chapter 7: Conclusions 3 2 Introduction 5 2.1 Background 5
2.2 Relationship Between Error and Uncertainty 7 2.3 Expert Perspectives on
Interval Analysis 8 2.4 Precision Analysis of Interval-Based Models
(Self-Validated Numeric) 10 2.5 Concepts of the Ordinary and Interval-based
Optimal Control 11 2.6 Orthogonal Spectral Methods 13 2.7 Desired
Confidence Interval 13 2.8 Conclusions 14 3 Literature Review 17 4
Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 29 4.1 Introduction 29 4.2 Introduction to IA 30 4.3 The
Algebra of Interval Sets 31 4.3.1 Classic Interval Algebra (Minkowski
Method) 31 4.3.2 Mathematical Norm and Distance in the IA 33 4.3.3 Kaucher
Extended Interval Analysis 33 4.3.4 Modal Interval Analysis 34 4.3.5
Hukuhara Difference Method 35 4.4 Interval Representations 36 4.5 Interval
Complex Integers 39 4.6 Interval Vector and Matrix 41 4.6.1 Vector and
Matrix IA 41 4.6.2 Interval Vector Norm 42 4.6.3 Interval Matrix Analysis
42 4.6.3.1 Conceptional Example: Wrapping Effect 43 4.6.4 Interval Matrices
Norm 44 4.6.5 Interval Determination of a Matrix 45 4.6.6 The Inverse of a
Regular Interval Matrix 45 4.6.7 Eigenvalues and Eigenvectors of an
Interval Matrix 45 4.7 Solving Linear Systems with Interval Parameters 46
4.8 Interval Functions 46 4.8.1 Overestimated Interval Function 46 4.8.2
Minimal Interval Function 48 4.9 Determining the Minimal Interval 49 4.9.1
Uniform Interval Functions 49 4.9.2 Nonuniform Interval Functions 49 4.9.3
Interval Power Series 50 4.10 Interval Derivative and Integral Functions 52
4.10.1 Interval Derivative 52 4.10.2 Interval Integration 53 4.11 Centered
Inclusion Method 54 4.11.1 Linearized Interval Functions Around the Center
54 4.11.2 Taylor Inclusion Functions 54 4.12 Interval Nonlinear Systems 56
4.13 Analysis of the Interval Dynamic Systems in the Presence of Interval
Uncertainties 57 4.13.1 Solving the Interval Initial Value Problems 58 4.14
The Interval Runge-Kutta Method (IRKM) for Interval Differential Equations
60 4.14.1 Introduction 60 4.14.2 Generalized IRKM (GIRKM) Based on
Switching Points 63 4.14.3 Numerical Examples 66 4.15 Interval Uncertainty
Analyses based on Orthogonal Functions 78 4.15.1 Interval
Epsilon-orthogonal 79 4.15.2 Interval Weierstrass's Theorem 79 4.16
Interval Orthogonal Polynomials 79 4.16.1 Legendre Polynomials 80 4.16.2
Chebyshev Polynomials 80 4.16.3 Interval Orthogonal Functions 82 4.17
Piecewise Extension of the Interval Orthogonal Functions 83 4.18 Conclusion
85 5 Stability and Controllability Based on Interval Analysis 91 5.1
Introduction 91 5.1.1 Classical Control Theory 91 5.1.2 Advanced Modern
Control Systems Theory 92 5.1.3 Optimal Control 93 5.1.4 Robust Control 94
5.1.5 Adaptive Control Theory 95 5.2 Interval Stability and Controllability
95 5.3 Interval Stability 96 5.4 Characteristic Polynomial 97 5.5
Routh-Hurwitz Stability Test 98 5.6 Kharitonov's Theorem (Interval
Routh-Hurwitz Stability Test) 99 5.6.1 Kharitonov Polynomial Theory 99
5.6.2 A Centered Representation of the Interval Routh-Hurwitz Stability
Criterion 102 5.7 Interval Stability Based on Linear Matrix Inequalities
102 5.7.1 The Positive Matrix of the Interval Matrix 102 5.7.2 Stability
Analysis of the Interval Systems 103 5.7.3 Linear Matrix Inequalities 104
5.8 Controllability and Observability 107 5.9 Controllability and
Observability Based on Interval Criteria 107 5.9.1 Singular Values for
Analyzing the Controllability 111 5.10 Conclusions 117 6 Optimal Control of
the Systems with Interval Uncertainties 121 6.1 Introduction 121 6.2
Indirect Methods 123 6.3 Direct Methods 123 6.4 Optimal Control Problem in
the Presence of Interval Uncertainties 127 6.5 Interval Optimal Control
Based on the Indirect Method 128 6.5.1 Analysis of the Standard Interval
Calculus of Variations 129 6.6 Analysis of the Problem of the Interval
Optimal Control Based on Euler-Lagrange Equations 131 6.7 Solving Optimal
Control Problems with Interval Uncertainties: Interval Runge-Kutta Method
132 6.8 Optimal Control of Problems with Interval Uncertainties Using the
Chebyshev Inclusion Method 146 6.9 Piecewise Interval Chebyshev Method for
OCPs 152 6.10 Solving Quadratic Optimal Control Problems with Interval
Uncertainties Based on Indirect Method: Interval Quadratic Regulator 159
6.11 Problem Statement (Interval Quadratic Regulator) 173 6.12 Interval
Optimal Control Based on Direct Method 179 6.13 Applied Simulations 183
6.14 Conclusion 192 References 193 7 Conclusions 197 Index 199
About the Author viii 1 Preface and Overview 1 Chapter 1: Preface and
Overview 1 Chapter 2: Introduction 2 Chapter 3: Literature Review 2 Chapter
4: Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 2 Chapter 5: Stability and Controllability Based on Interval
Analysis 3 Chapter 6: Optimal Control of the Systems with Interval
Uncertainties 3 Chapter 7: Conclusions 3 2 Introduction 5 2.1 Background 5
2.2 Relationship Between Error and Uncertainty 7 2.3 Expert Perspectives on
Interval Analysis 8 2.4 Precision Analysis of Interval-Based Models
(Self-Validated Numeric) 10 2.5 Concepts of the Ordinary and Interval-based
Optimal Control 11 2.6 Orthogonal Spectral Methods 13 2.7 Desired
Confidence Interval 13 2.8 Conclusions 14 3 Literature Review 17 4
Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 29 4.1 Introduction 29 4.2 Introduction to IA 30 4.3 The
Algebra of Interval Sets 31 4.3.1 Classic Interval Algebra (Minkowski
Method) 31 4.3.2 Mathematical Norm and Distance in the IA 33 4.3.3 Kaucher
Extended Interval Analysis 33 4.3.4 Modal Interval Analysis 34 4.3.5
Hukuhara Difference Method 35 4.4 Interval Representations 36 4.5 Interval
Complex Integers 39 4.6 Interval Vector and Matrix 41 4.6.1 Vector and
Matrix IA 41 4.6.2 Interval Vector Norm 42 4.6.3 Interval Matrix Analysis
42 4.6.3.1 Conceptional Example: Wrapping Effect 43 4.6.4 Interval Matrices
Norm 44 4.6.5 Interval Determination of a Matrix 45 4.6.6 The Inverse of a
Regular Interval Matrix 45 4.6.7 Eigenvalues and Eigenvectors of an
Interval Matrix 45 4.7 Solving Linear Systems with Interval Parameters 46
4.8 Interval Functions 46 4.8.1 Overestimated Interval Function 46 4.8.2
Minimal Interval Function 48 4.9 Determining the Minimal Interval 49 4.9.1
Uniform Interval Functions 49 4.9.2 Nonuniform Interval Functions 49 4.9.3
Interval Power Series 50 4.10 Interval Derivative and Integral Functions 52
4.10.1 Interval Derivative 52 4.10.2 Interval Integration 53 4.11 Centered
Inclusion Method 54 4.11.1 Linearized Interval Functions Around the Center
54 4.11.2 Taylor Inclusion Functions 54 4.12 Interval Nonlinear Systems 56
4.13 Analysis of the Interval Dynamic Systems in the Presence of Interval
Uncertainties 57 4.13.1 Solving the Interval Initial Value Problems 58 4.14
The Interval Runge-Kutta Method (IRKM) for Interval Differential Equations
60 4.14.1 Introduction 60 4.14.2 Generalized IRKM (GIRKM) Based on
Switching Points 63 4.14.3 Numerical Examples 66 4.15 Interval Uncertainty
Analyses based on Orthogonal Functions 78 4.15.1 Interval
Epsilon-orthogonal 79 4.15.2 Interval Weierstrass's Theorem 79 4.16
Interval Orthogonal Polynomials 79 4.16.1 Legendre Polynomials 80 4.16.2
Chebyshev Polynomials 80 4.16.3 Interval Orthogonal Functions 82 4.17
Piecewise Extension of the Interval Orthogonal Functions 83 4.18 Conclusion
85 5 Stability and Controllability Based on Interval Analysis 91 5.1
Introduction 91 5.1.1 Classical Control Theory 91 5.1.2 Advanced Modern
Control Systems Theory 92 5.1.3 Optimal Control 93 5.1.4 Robust Control 94
5.1.5 Adaptive Control Theory 95 5.2 Interval Stability and Controllability
95 5.3 Interval Stability 96 5.4 Characteristic Polynomial 97 5.5
Routh-Hurwitz Stability Test 98 5.6 Kharitonov's Theorem (Interval
Routh-Hurwitz Stability Test) 99 5.6.1 Kharitonov Polynomial Theory 99
5.6.2 A Centered Representation of the Interval Routh-Hurwitz Stability
Criterion 102 5.7 Interval Stability Based on Linear Matrix Inequalities
102 5.7.1 The Positive Matrix of the Interval Matrix 102 5.7.2 Stability
Analysis of the Interval Systems 103 5.7.3 Linear Matrix Inequalities 104
5.8 Controllability and Observability 107 5.9 Controllability and
Observability Based on Interval Criteria 107 5.9.1 Singular Values for
Analyzing the Controllability 111 5.10 Conclusions 117 6 Optimal Control of
the Systems with Interval Uncertainties 121 6.1 Introduction 121 6.2
Indirect Methods 123 6.3 Direct Methods 123 6.4 Optimal Control Problem in
the Presence of Interval Uncertainties 127 6.5 Interval Optimal Control
Based on the Indirect Method 128 6.5.1 Analysis of the Standard Interval
Calculus of Variations 129 6.6 Analysis of the Problem of the Interval
Optimal Control Based on Euler-Lagrange Equations 131 6.7 Solving Optimal
Control Problems with Interval Uncertainties: Interval Runge-Kutta Method
132 6.8 Optimal Control of Problems with Interval Uncertainties Using the
Chebyshev Inclusion Method 146 6.9 Piecewise Interval Chebyshev Method for
OCPs 152 6.10 Solving Quadratic Optimal Control Problems with Interval
Uncertainties Based on Indirect Method: Interval Quadratic Regulator 159
6.11 Problem Statement (Interval Quadratic Regulator) 173 6.12 Interval
Optimal Control Based on Direct Method 179 6.13 Applied Simulations 183
6.14 Conclusion 192 References 193 7 Conclusions 197 Index 199
Overview 1 Chapter 2: Introduction 2 Chapter 3: Literature Review 2 Chapter
4: Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 2 Chapter 5: Stability and Controllability Based on Interval
Analysis 3 Chapter 6: Optimal Control of the Systems with Interval
Uncertainties 3 Chapter 7: Conclusions 3 2 Introduction 5 2.1 Background 5
2.2 Relationship Between Error and Uncertainty 7 2.3 Expert Perspectives on
Interval Analysis 8 2.4 Precision Analysis of Interval-Based Models
(Self-Validated Numeric) 10 2.5 Concepts of the Ordinary and Interval-based
Optimal Control 11 2.6 Orthogonal Spectral Methods 13 2.7 Desired
Confidence Interval 13 2.8 Conclusions 14 3 Literature Review 17 4
Introduction to Interval Analysis and Solving the Problems with Interval
Uncertainties 29 4.1 Introduction 29 4.2 Introduction to IA 30 4.3 The
Algebra of Interval Sets 31 4.3.1 Classic Interval Algebra (Minkowski
Method) 31 4.3.2 Mathematical Norm and Distance in the IA 33 4.3.3 Kaucher
Extended Interval Analysis 33 4.3.4 Modal Interval Analysis 34 4.3.5
Hukuhara Difference Method 35 4.4 Interval Representations 36 4.5 Interval
Complex Integers 39 4.6 Interval Vector and Matrix 41 4.6.1 Vector and
Matrix IA 41 4.6.2 Interval Vector Norm 42 4.6.3 Interval Matrix Analysis
42 4.6.3.1 Conceptional Example: Wrapping Effect 43 4.6.4 Interval Matrices
Norm 44 4.6.5 Interval Determination of a Matrix 45 4.6.6 The Inverse of a
Regular Interval Matrix 45 4.6.7 Eigenvalues and Eigenvectors of an
Interval Matrix 45 4.7 Solving Linear Systems with Interval Parameters 46
4.8 Interval Functions 46 4.8.1 Overestimated Interval Function 46 4.8.2
Minimal Interval Function 48 4.9 Determining the Minimal Interval 49 4.9.1
Uniform Interval Functions 49 4.9.2 Nonuniform Interval Functions 49 4.9.3
Interval Power Series 50 4.10 Interval Derivative and Integral Functions 52
4.10.1 Interval Derivative 52 4.10.2 Interval Integration 53 4.11 Centered
Inclusion Method 54 4.11.1 Linearized Interval Functions Around the Center
54 4.11.2 Taylor Inclusion Functions 54 4.12 Interval Nonlinear Systems 56
4.13 Analysis of the Interval Dynamic Systems in the Presence of Interval
Uncertainties 57 4.13.1 Solving the Interval Initial Value Problems 58 4.14
The Interval Runge-Kutta Method (IRKM) for Interval Differential Equations
60 4.14.1 Introduction 60 4.14.2 Generalized IRKM (GIRKM) Based on
Switching Points 63 4.14.3 Numerical Examples 66 4.15 Interval Uncertainty
Analyses based on Orthogonal Functions 78 4.15.1 Interval
Epsilon-orthogonal 79 4.15.2 Interval Weierstrass's Theorem 79 4.16
Interval Orthogonal Polynomials 79 4.16.1 Legendre Polynomials 80 4.16.2
Chebyshev Polynomials 80 4.16.3 Interval Orthogonal Functions 82 4.17
Piecewise Extension of the Interval Orthogonal Functions 83 4.18 Conclusion
85 5 Stability and Controllability Based on Interval Analysis 91 5.1
Introduction 91 5.1.1 Classical Control Theory 91 5.1.2 Advanced Modern
Control Systems Theory 92 5.1.3 Optimal Control 93 5.1.4 Robust Control 94
5.1.5 Adaptive Control Theory 95 5.2 Interval Stability and Controllability
95 5.3 Interval Stability 96 5.4 Characteristic Polynomial 97 5.5
Routh-Hurwitz Stability Test 98 5.6 Kharitonov's Theorem (Interval
Routh-Hurwitz Stability Test) 99 5.6.1 Kharitonov Polynomial Theory 99
5.6.2 A Centered Representation of the Interval Routh-Hurwitz Stability
Criterion 102 5.7 Interval Stability Based on Linear Matrix Inequalities
102 5.7.1 The Positive Matrix of the Interval Matrix 102 5.7.2 Stability
Analysis of the Interval Systems 103 5.7.3 Linear Matrix Inequalities 104
5.8 Controllability and Observability 107 5.9 Controllability and
Observability Based on Interval Criteria 107 5.9.1 Singular Values for
Analyzing the Controllability 111 5.10 Conclusions 117 6 Optimal Control of
the Systems with Interval Uncertainties 121 6.1 Introduction 121 6.2
Indirect Methods 123 6.3 Direct Methods 123 6.4 Optimal Control Problem in
the Presence of Interval Uncertainties 127 6.5 Interval Optimal Control
Based on the Indirect Method 128 6.5.1 Analysis of the Standard Interval
Calculus of Variations 129 6.6 Analysis of the Problem of the Interval
Optimal Control Based on Euler-Lagrange Equations 131 6.7 Solving Optimal
Control Problems with Interval Uncertainties: Interval Runge-Kutta Method
132 6.8 Optimal Control of Problems with Interval Uncertainties Using the
Chebyshev Inclusion Method 146 6.9 Piecewise Interval Chebyshev Method for
OCPs 152 6.10 Solving Quadratic Optimal Control Problems with Interval
Uncertainties Based on Indirect Method: Interval Quadratic Regulator 159
6.11 Problem Statement (Interval Quadratic Regulator) 173 6.12 Interval
Optimal Control Based on Direct Method 179 6.13 Applied Simulations 183
6.14 Conclusion 192 References 193 7 Conclusions 197 Index 199