iosr-Jm   Volume-9 ~ Issue-5

Abstract: This paper investigates the application of the iterative Steepest Descent Method (SDM) to the solution of nonlinear two-point boundary-value problems for the optimal controls and trajectories of continuous-time linear-quadratic regulator problems. Numerical results show some improvement over the classical methods.

Keywords: Optimal Control, Continuous Linear Regulator, Steepest Descent Method..

[1] Athans, M. and Falb, P. L., (1966), Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
[2] Bryson, A. E. Jr., and W. F., Denham, (1964), "Optimal Programming Problems with Inequality Constraints II: Solution by Steepest Ascent," AIAA Journal, 25-34.
[3] Burghes, D. N. and Graham, A., (1980), Introduction To Control Theory, Including Optimal Control, John Wiley & Sons.
[4] George M. Siouris, (1996), An Engineering Approach To Optimal Control And Estimation Theory, John Wiley & Sons, Inc.
[5] Kelly, H. J., (1962), "Method of Gradient," Optimization Techniques with Applications To Aerospace Systems. G. Leitmann, Ed. New York: Academic Press Inc.

Abstract: This paper investigates and discusses the method of dynamic programming in solving Bolza's cost form of Linear Quadratic Regulator Problems (LQRP). It is the desire of the authors of this paper to experiment numerically the solution of this class of problem using dynamic programming to solve for the optimal controls and the trajectories compared with other numerical methods with a view to further improving the results. The method uses the principle of optimality to reduce mathematically the number of calculations required to determine the optimal control law as well as the corresponding optimal cost functional.

Keywords: Continuous-Time Linear Regulator Problem, Optimal Control, Discretization and Dynamic Programming.

[1] Boudarel, J., et al, (1971), Dynamic Programming and Its Application to Optimization Theory, Academic Press, Inc., 111 Fifth Avenue, New York, New York 10003.
[2] David, G. Hull, (2003), Optimal Control Theory for Applications, Mechanical Engineering Series, Springer-Verlag, New York, Inc., 175 Fifth Avenue, New York, NY 10010.
[3] Kirk, E. Donald, (2004), Optimal control theory: An introduction, Prentice-Hall, Inc., Englewood Cliff, New Jersey.
[4] George M. Siouris, (1996), An Engineering Approach To Optimal Control And Estimation Theory, John Wiley & Sons, Inc., 605 Third Avenue, New York, 10158- 00 12.

Abstract:The present paper deals with Latin squares, orthogonal Latin squares, mutually Orthogonal Latin squares, close connections between Latin squares and finite geometries. Moreover the great mathematician Leonhard Euler introduced Latin squares in 1783 as a "nouveau espece de carres magiques", a new kind of magic squares. He also defined what he meant by orthogonal Latin squares, which led to a famous conjecture of his that went unsolved for over 100 years.

[1] Mann, H. B., Analysis and Design of Experiments, Dover, New York, 1949.

[2] Denes, J. and A. D. Keedwell, Latin Squares and their Applications, Academic Press, New York, 1974.

[3] Bose, Ray Chandra, "On the Application of the Properties of Galois Fields to the Problem of Construction of Hyper-Graeco-Latin Squares," Sankhya (The Indian Journal of Statistics), 3, 323-338(1938).

[4] Ball, W. W. Rouse, Mathematical Recreations and Essays, rev. ed. Macmillan, London, 1939.

Abstract: In this paper, we constructed a control operator sequel to an earlier constructed control operator in one of our papers which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving discrete time linear quadratic regulator problems with delay parameter in the state variable. The construction of the control operator places scalar linear delay problems of the type within the class of problems that can be solved with the ECGM and it is aimed at reducing the rigours faced in using the classical methods in solving this class of problem.

[1] ADEBAYO,K. J. and ADERIBIGBE, F. M., (2014), On Construction of A Control Operator Applied In Conjugate Gradient Method Algorithm For Solving Continuous Time Linear Regulator Problems With Delay – I
[2] Aderibigbe, F. M., (1993), "An Extended Conjugate Gradient Method Algorithm For Control Systems with Delay-I, Advances in Modeling & Analysis, C, AMSE Press, Vol. 36, No. 3, pp 51-64.
[3] Athans, M. and Falb, P. L., (1966), Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
[4] David, G. Hull, (2003), Optimal Control Theory for Applications, Mechanical Engineering Series, Springer-Verlag, New York, Inc., 175 Fifth Avenue, New York, NY 10010.

Abstract: In this paper the terms, simple po ternary semigroup, globally idempotent po ternary semigroup, semisimple element, regular element, left regular element, lateral regular element, right regular element, completely regular element, intra regular element in a poternary semigroup are introduced. It is proved that, if a is a completely regular element of a po ternary semigroup T then a is left regular, lateral regular and right regular. It is proved that in any po ternary semigroup T, the following are equivalent (1) Principal po ideals of T form a chain. (2) Po ideals of T form a chain. It is proved that a po ternary semigroup T is simple po ternary semigroup if and only if ( TTaTT ] = T for all aT.

[1] Anjaneyulu. A .,Structure and ideal theory of Duo semigroups, Semigroup forum, vol .22(1981), 237-276.
[2] Clifford A.H and Preston G.B., The algebroic theory of semigroups , vol – I American Math. Society, Province (1961).
[3] Clifford A.H and Preston G.B., The algebroic theory of semigroups , vol – II American Math. Society, Province (1967).
[4] Hewitt.E. and Zuckerman H.S., Ternary opertions and semigroups, semigroups, Proc. Sympos. Wayne State Univ., Detroit,
1968,33-83.
[5] Iampan . A., Lateral ideals of ternary semigroups , Ukrainian Math, Bull., 4 (2007), 323-334.

Abstract: This paper investigates and discusses modification on the gradient of continuous function input introduced to Extended Conjugate Gradient Method algorithm (ECGM) employed in solving optimal control problems with the state variable constrained by a differential equation. Numerical results show some improvement over the classical methods. .

Keywords: Optimal Control, Control Operator, Conjugate Gradient Method and Extended Conjugate Gradient Method.

1] Aderibigbe, F. M.: "An Extended Conjugate Gradient Algorithm for Control Systems with Delay-I", Advances in Modelling & Analysis, CAMSE Press, Vol. 36, No. 3, (1993), 51-64.
[2] Aderibigbe, F. M. and Adebayo, K. J.: "A Scaled Extended Conjugate Gradient Method", Jour. of Mathematical Sciences, Vol. 23, No. 1, (2011), 189-196.
[3] George M. Siouris,: An Engineering Approach To Optimal Control And Estimation Theory, John Wiley & Sons, Inc., (1996)
[4] Hasdorff, L., Gradient Optimization and Nonlinear Control, J. Wiley and Sons, New York, (1976).
[5] Kirk, E. Donald,: Optimal Control Theory, An Introduction, Prentice-Hall, Inc. Englewood Cliffs New Jersey, (2004)
[6] Ibiejugba, M. A. and Onumanyi, P.: "Control Operator and Some of Its Applications", J. Math. Anal. Appl. Vol. 103, No. 1, (1984), 31-47.
[7] Otunta, F. O. and Apanapudor, J. S.,: "The Construction of A Hybrid Operator ̃", Jour. of Inst. of Maths. & Comp. Sciences, Vol. 21, No. 2, (2008), 61-71.
[8] Emmanuel, N.,: "An Extended Conjugate Gradient Method for Solving Equality Constrained Nonlinear Optimization Problems", Canadian Jour. on Computing in Maths, Natural Sciences, Engineering & Medicine, Vol. 2, No. 3, (2011), 45-49.

Abstract: In this paper, we constructed a control operator, G, which enables a Conjugate Gradient Method (CGM) to be employed in solving continuous time linear regulator problems with delay parameter in the state variable. The control operator takes care of any of Mayer's, Lagrange's and Bolza's cost form of linear regulator problems. It is the desire of the authors of this paper that the application of this control operator will further improve on the result of the Conjugate Gradient Method in solving this class of optimal control problem.

[1] Athans, M. and Falb, P. L., (1966), Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
[2] Aderibigbe, F. M., (1993), "An Extended Conjugate Gradient Method Algorithm For Control Systems with Delay-I, Advances in Modeling & Analysis, C, AMSE Press, Vol. 36, No. 3, pp 51-64.
[3] George M. Siouris, (1996), An Engineering Approach To Optimal Control And Estimation Theory, John Wiley & Sons, Inc., 605 Third Avenue, New York, 10158-00 12.
[4] Hasdorff, L. (1976), Gradient optimization and Nonlinear Control. J. Wiley and Sons, New York.
[5] Ibiejugba, M. R. and Onumanyi, P., (1984), "A Control Operator and some of its Applications, J. Math. Anal. Appl. Vol. 103, No. 1, Pp. 31-47.

Abstract:This article deals with the dynamic transient thermal stresses in a solid sphere of a functionally graded material. The sphere material is considered to be graded along the radial direction, where an exponential varying distribution is assumed. The Poisson's ratio assumed to be constant. The sphere is subjected to a constant temperature at the circular surface of sphere. A numerical finite difference method is used to obtain the time dependent temperature, displacement and thermal stress distribution and results are presented for the FGM sphere consisting ZrO2 and Ti-6A1-4V.

Keywords: Transient Thermal stresses, functionally graded material

[1] Noda N., Hetnarski R.B.and Tanigawa Y., Thermal Stresses, Taylor and Francis, New York, 302, 2nd Ed: (2003).
[2] Obata Y. and Noda N., Steady State Thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded
material, J. Thermal Stresses,17(3), 471- 487, (1994).
[3] Ootao Y., Akai T. and Tanigawa Y., Three dimensional Transient thermal stresses Analysis of a non-homogeneous hollow circular
cylinder due to moving heat source in the axial direction , J. Thermal stresses vol. 18, 497-512,(1995)
[4] Lutz M. P., Zimmerman R. W., Thermal stresses and effective thermal expansion coefficient of A Functionally Graded Sphere, J.
Thermal Stresses,19,39-54, (1996)

Abstract: It was shown in Sadiq (2013) that succession parameters under the Aunu permutation patterns can be used as vertices of the graph model resulting from different transition of the automata scheme employed. This paper generates a graph model using the Aunu permutation patterns governed by some properties as embedded in method of construction of a typical game of chance scheme.

1]. Ibrahim, A. A. (2004), "On wreath product of permutation Groups and algebraic theoretic properties of bara'at al-dhimmah models," PhD Thesis.
[2]. Ibrahim, A. A. (2005) "On the combination of succession in it – 5 element sample,"Abacus journal of mathematics Association on Nigeria Vol. 32 No. 2B 410-415
[3]. Ibrahim, A.A. and Audu, M.S. (2005),"Some group theoretic properties of certain class of (123) and (132) avoiding patterns of certain numbers; An enumeration scheme,"African Journal of Natural Science. Vol. 8:79 – 84
[4]. Ibrahim, A.A. and Audu, M.S. (2008), "A stable variety of Cayley Graphs for Efficient Interconnection networks,"proceeding of Annual conference of mathematical Association of Nigeria, 156 – 161.
[5]. Ibrahim, A.A. and M.S. Audu, (2008), "On stable variety of Cayley Graphs for Efficient interconnection networks," proceedings of Annual National conference of mathematical Association of Nigeria (MAN),156 – 161.
[6]. Jeffrey, D.Ullman, (2000),Introduction to Automata theory, languages and computation, Second Edition. 32: 456-510.
[7]. Kathleen, (2001). "Theory and Application of multiple Attractor Cellular Automata for fauit Diagnosis," international journal of Asian test Symposium. 321:388-392.
[8]. Sadiq S, et-al. (2013). Construction of finite Cellular Automata using (123)- avoiding Class of the Aunu Permutation Patterns: Application in Game of Life

Abstract: We prove a unique common fixed-point theorem for two pair of weakly compatible maps in a complete metric space, which generalizes the result of Brian Fisher by a weaker condition such as weakly compatibility instead of compatibility and contractive modulus instead of continuity of maps.

Keywords: Fixed point, Common fixed point, Contractive modulus, weakly compatible maps, and complete metric space.

[1] Aamri, M. and El Moutawakil, D.: some new common fixed theorems under strict conditions, J. Math. Anal. Appl., 270(2002), 181-188.
[2] Fisher, B.: Common Fixed Point of Four Mapping. Bull. Inst. Of Math. Academia. Sinicia. 11(1983), 103-113.
[3] Imdad, M. Kumar, Santosh. And Khan, M. S.:Remarks on fixed Point theorems satisfying implicit relations, Radovi Math., 11(2002), 135-143.
[4] Jungck, G.: Compatible mappings and common fixed points, internet. I. Math and Math. Sci., 9(1986), 771-779.
[5] Jungck, G.and Rhoades, B.E.: Fixed Point for set valued Functions Without Continuty, Indian J. Pure Appl. Math, 29(1998), 227-238.