## Table of Contents

**Table of Contents**

Preface

Chapter 1. Introduction on Cutting Angle Method Inspired By Abstract Convexity for Solving Continuous Time Optimal Control Problems

Chapter 2. A Wide Literature Review on Building the Cutting Angle Method

Chapter 3. The Inheritance and Generalizability Properties Extended From Function Definitions into Functionals

Chapter 4. Study of Some New Type of Functionals Defined Based the Inheritance and Generalizability Properties of Functions

Chapter 5. Capability of Function Optimization Algorithms for Solving Optimal Control Problems With Respect to the Inheritance and Generalizability Properties

Chapter 6. A Generalized Version of Cutting Angle Method for Solving Continuous Time Optimal Control Problems

Chapter 7. A Combination of the Cutting Angle Method and a Local Search on Optimal Control Problems

Index

**References**

Chapter 1

[1] Baldi, P. (1995). â€śGradient descent learning algorithm overview: A general dynamical systems perspectiveâ€ť, Neural Networks, IEEE Transactions on, 6(1), 182-195.

[2] Shang, Y. (1997). â€śGlobal search methods for solving nonlinear optimization problems,â€ť University of Illinois at Urbana-Champaign.

[3] Beale, E. M. L. (1988). Introduction to optimization: Wiley-Interscience.

[4] Bundy, B. (1984). â€śBasic optimization methodsâ€ť, Edward Arnold.

[5] Cesari, L. (1983). Optimizationâ€”theory and applications: Springer.

[6] Koo, D. (1977). Elements of optimization: Springer.

[7] Rao, S. â€śOptimization: theory and applications. 1984,â€ť ed: Wiley, New York.

[8] Stoer, J. & Witzgall, C. (1970). Convexity and optimization in finite dimensions: Springer.

[9] Rubinov, A. M. (2000). Abstract convexity and global optimization, vol. 44, Springer.

[10] Kurdila, A. J. & Zabarankin, M. (2005). â€śConvex Functional Analysis, Systems and Control: Foundations and Applications,â€ť ed: BirkhĂ¤user, Basel.

[11] Nocedal, J. & Wright, S. J. (2006). â€śNumerical Optimization 2ndâ€ť.

[12] Rubinov, A. & Glover, B. (1999). â€śIncreasing convex-along-rays functions with applications to global optimizationâ€ť, Journal of Optimization Theory and Applications, 102(3), 615-642.

[13] Singh, M. G. & Titli, A. (1987). Systems: decomposition, optimisation, and control: Pergamon.

[14] Becerra, V. M. (1994). â€śDevelopment and applications of novel optimal control algorithms,â€ť City University.

[15] Roberts, P. (1993). â€śAn algorithm for optimal control of nonlinear systems with model-reality differencesâ€ť, in Proceedings of 12th IFAC World Congress on Automatic Control, 407-412.

[16] Bagirov, A. & Rubinov, A. (2000). â€śGlobal minimization of increasing positively homogeneous functions over the unit simplexâ€ť, Annals of Operations Research, 98(1-4), 171-187.

Chapter 2

[1] Martinez-Legaz, J. & Rubinov, A. (2001). â€śIncreasing positively homogeneous functions defined on Rnâ€ť, Acta Math. Vietnam, 26(3), 313-331.

[2] Fletcher, R. (2013). Practical methods of optimization: John Wiley & Sons.

[3] Rao, S. â€śOptimization: theory and applications. 1984,â€ť ed: Wiley, New York.

[4] Baldi, P. (1995). â€śGradient descent learning algorithm overview: A general dynamical systems perspectiveâ€ť, Neural Networks, IEEE Transactions on, 6(1), 182-195.

[5] Beale, E. M. L. (1988). Introduction to optimization: Wiley-Interscience.

[6] Bundy, B. (1984). â€śBasic optimization methodsâ€ť, Edward Arnold.

[7] Cesari, L. (1983). Optimizationâ€”theory and applications: Springer.

[8] Stoer, J. & Witzgall, C. (1970). Convexity and optimization in finite dimensions: Springer.

[9] Hansen, P. & Jaumard, B. (1995). Lipschitz optimization: Springer.

[10] Mladineo, R. H. (1986). â€śAn algorithm for finding the global maximum of a multimodal, multivariate functionâ€ť, Mathematical Programming, 34(2), 188-200.

[11] Bagirov, A. M. & Rubinov, A. M. (2003). â€śCutting angle method and a local searchâ€ť, Journal of Global Optimization, 27(2-3), 193-213.

[12] Siouris, G. M. (1996). An engineering approach to optimal control and estimation theory: John Wiley & Sons, Inc.

[13] Hocking, L. (2001). â€śOptimal control. An introduction to the theory with applications.â€ť Clarendon, ed: Oxford.

[14] BrdyĹ›, M. & Roberts, P. (1987). â€śConvergence and optimality of modified two-step algorithm for integrated system optimization and parameter estimationâ€ť, International journal of systems science, 18(7), 1305-1322.

[15] Ellis, J. & Roberts, P. (1981). â€śSimple models for integrated optimization and parameter estimationâ€ť, International Journal of Systems Science, 12(4), 455-472.

[16] Roberts, P. & Williams, T. (1981). â€śOn an algorithm for combined system optimisation and parameter estimationâ€ť, Automatica, 17(1), 199-209.

[17] Stevenson, I., Brdys, M. & Roberts, P. (1985). â€śIntegrated system optimization and parameter estimation for travelling load furnace controlâ€ť, in Proceedings of 7th IFAC Symposium on Identification and Parameter Estimation, 1641-1646.

[18] Becerra, V. & Roberts, P. (1996). â€śDynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systemsâ€ť, International Journal of Control, 63(2), 257-281.

[19] Roberts, P. (1993). â€śAn algorithm for optimal control of nonlinear systems with model-reality differencesâ€ť, in Proceedings of 12th IFAC World Congress on Automatic Control, 407-412.

[20] Becerra, V. & Roberts, P. (1998). â€śApplication of a novel optimal control algorithm to a benchmark fed-batch fermentation processâ€ť, Transactions of the Institute of Measurement and Control, 20(1), 11-18.

Chapter 3

[1] Munkres, J. R. (2000). â€śTopology Prentice Hallâ€ť, Upper Saddle River, NJ.

[2] Barbu, V. & Precupanu, T. (2012). Convexity and optimization in Banach spaces: Springer.

[3] Roekafellar, R. (1970). â€śConvex analysisâ€ť, Princeton.

[4] Kudryavtsev, L. (2002). â€śHomogeneous functionâ€ť, Encyclopaedia of mathematics. Springer, Berlin.

[5] Kurdila, A. J. & Zabarankin, M. (2005). â€śConvex Functional Analysis, Systems and Control: Foundations and Applications,â€ť ed: BirkhĂ¤user, Basel.

[6] Sloughter, D. (2001). â€śThe Calculus of Functions of Several Variables,â€ť ed: Furman University, 260p.

[7] Boyd, S. P., El Ghaoui, L., Feron, E. & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory, vol. 15, SIAM.

[8] Bittner, L. (1974). â€śOn moment equations and inequalitiesâ€ť, Lithuanian Mathematical Journal, 14(2), 178-185.

[9] Castagnoli, E. & Maccheroni, F. (2000). â€śRestricting independence to convex conesâ€ť, Journal of Mathematical Economics, 34(2), 215-223.

Chapter 4

[1] Bylinski, C. (1989). â€śSome basic properties of setsâ€ť, Journal of Formalized Mathematics, 1(198), 9.

[2] Hryniewiecki, K. (1990). â€śBasic properties of real numbersâ€ť, Formalized Mathematics, 1(1), 35-40.

[3] Kotowicz, J. & Sakai, Y. (1992). â€śProperties of partial functions from a domain to the set of real numbersâ€ť, Formalized Mathematics, 3(2), 279-288.

[4] Nowak, B. & Trybulec, A. (1993). â€śHahn-Banach theoremâ€ť, Journal of Formalized Mathematics, 5(199), 3.

[5] Hamel, A. (2004). â€śFrom real to set-valued coherent risk measuresâ€ť, Reports of the Institute of Optimization and Stochastics, Martin-Luther-University Halle-Wittenberg, 19, 10-22.

[6] Anger, B. (1977). â€śRepresentation of capacitiesâ€ť, Mathematische Annalen, 229(3), 245-258.

[7] Kurdila, A. J. & Zabarankin, M. (2005). â€śConvex Functional Analysis, Systems and Control: Foundations and Applications,â€ť ed: BirkhĂ¤user, Basel.

[8] Koliha, J. J. & Leung, A. (1975). â€śErgodic families of affine operatorsâ€ť, Mathematische Annalen, 216(3), 273-284.

[9] Rubinov, A. M. (2000). Abstract convexity and global optimization vol. 44, Springer.

Chapter 5

[1] Becerra, V. M. (2008). â€śOptimal controlâ€ť, Scholarpedia, 3(1), 5354.

[2] Seyedi, S., Ahmad, R. & Aziz, M. I. A. (2009). â€śInheritance of Function Properties for Functionalsâ€ť.

[3] Kazemi, M. & Miri, M. (1993). â€śNumerical solution of optimal control problemsâ€ť, in Southeastconâ€™93, Proceedings, IEEE, 2 p.

[4] Fard, O. S. & Borzabadi, A. H. (2007). â€śOptimal control problem, quasi-assignment problem and genetic algorithmâ€ť, in Proceedings of World Academy of Science, Engineering and Technology, 70-43.

Chapter 6

[1] Singh, M. G. & Titli, A. (1978). Systems: decomposition, optimisation, and control: Pergamon.

[2] Becerra, V. M. (1994). â€śDevelopment and applications of novel optimal control algorithms,â€ť City University.

[3] Roberts, P. (1993), â€śAn algorithm for optimal control of nonlinear systems with model-reality differencesâ€ť, in Proceedings of 12th IFAC World Congress on Automatic Control, 407-412.

[4] Leithold, L. (1971). The Calculus Book.

[5] Fard, O. S. & Borzabadi, A. H. (2007). â€śOptimal control problem, quasi-assignment problem and genetic algorithmâ€ť, in Proceedings of World Academy of Science, Engineering and Technology, 70-43.

Chapter 7

[1] Bagirov, A. & Rubinov, A. (2000). â€śGlobal minimization of increasing positively homogeneous functions over the unit simplexâ€ť, Annals of Operations Research, 98(1-4), 171-187.

[2] Bagirov, A. M. & Rubinov, A. M. (2003). â€śCutting angle method and a local searchâ€ť, Journal of Global Optimization, 27(2-3), 193-213.

[3] Roberts, P. (1999). â€śStability properties of an iterative optimal control algorithmâ€ť, in Proceedings of 14th IFAC World Congress on Automatic Control, 269-274.

[4] Bryson, Jr. A. E. (1996) â€śOptimal control-1950 to 1985â€ť, Control Systems, IEEE, 16(3), 26-33.