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1. Department of Mathematics, Faculty of Sciences and Arts, ?anakkale Onsekiz Mart University, ?anakkale 17100, Turkey
2. Department of Mathematics, Faculty of Sciences and Letters, ?stanbul Technical University, ?stanbul 34469, Turkey
3. Department of Mathematics, Faculty of Sciences and Arts, Nam?k Kemal University, Tekirda? 59030, Turkey
Received: September 26, 2011 / Accepted: October 11, 2011 / Published: January 25, 2012.
Abstract: A process represented by nonlinear multi-parametric binary dynamic system is investigated in this work. This process is characterized by the pseudo Boolean objective functional. Since the transfer functions on the process are Boolean functions, the optimal control problem related to the process can be solved by relating between the transfer functions and the objective functional. An analogue of Bellman function for the optimal control problem mentioned is defined and consequently suitable Bellman equation is constructed.
Key words: Bellman equation, bellman function, galois field, shift operator, nonlinear multi-parametric binary dynamic system.
1. Introduction
The utility of nonlinear multi-parametric binary dynamic system (NMBDS) in potential applications in connection with sequential machines, cellular automata and coding theory, the command of technical processes with the help of computers, modeling of the moving objects and simulation of the operation for their control devices, designs of contemporary evaluation systems and other modern engineering problems makes it more actual for research. In this context, an optimal process represented by NMBDS is investigated and Bellman equation interested is derived in this work. The work is organized as follows: Section 2 introduces the process represented by NMBDS. The optimal control problem is presented in section 3. Bellman equation interested is constructed in section 4. Finally, conclusions are given in section 5.
2. NMBDS
Generally, this dynamic system is defined as follows[1-2]:
3. Optimal Control Problem
Now, we can state the original problem represented by NMBDS as follows: In order for a given NMBDS to go from the known initial state ?? to any desired states ??????, to which we expect to access in Lsteps, a control???????[6] must exist such that the functional in (4) has a minimal value:
Since the transfer functions are Boolean and the objective functional which characterizes the process is pseudo Boolean [3, 5], the pseudo Boolean expressions of the transfer functions have been obtained by the operations given in Ref. [7]. After this step, the problem can be stated as follows:
4. Construction of Bellman Equation
For every fixed σandχ, let a function be corresponded to the optimal value of pseudo Boolean functional in the problem (13)-(16). We say that this function is the piecewise analogue of Bellman function[8-9] in the problem (9)-(12):
The initial condition for Bellman equation is given on the right-upper region of ?? and directly determined with the help of the following equality
The condition implying the existence of the unique solution of the system of Eq. (13) is given by Ref. [2],
5. Conclusions
The suitable Bellman equation for piecewise function corresponding to optimal value of the functional in optimal control problem is constructed. It is demonstrated that this piecewise function is the solution to Bellman equation.
References
[1] F. Scheid, Schaum’s Outline of Theory and Problems of Numerical Analysis, McGraw-Hill, Usa, 1988, p. 184.
[2] I. V. Gaishun, Completely Solvable Multidimensional Differential Equations, Nauka and Tekhnika, Minsk, 1983, p. 231.
[3] J.A. Anderson, Discrete Mathematics with Combinatorics, Prentice Hall, New Jersey, 2004, p. 45.
[4] B. Musayev, M. Alp, Functional Analysis, Balc? Publications, Kütahya, Turkey, 2000, p. 7.
[5] S.V. Yablonsky, Introduction to Discrete Mathematics, Mir Publishers, Moscow, 1989, p. 9.
[6] Y.H. Hac?, K. Ozen, Terminal control problem for processes represented by nonlinear multi parameter binary dynamic system, Control and Cybernetics 38 (2009) 625-633.
[7] Y. Hac?, Pseudo Boolean expressions of multi-parametric Boolean vector transfer functions, in: Proceedings of II. Turkish World Mathematics Symposium, Sakarya, Turkey, 2007, p. 151.
[8] V.G. Boltyanskii, Optimal Control of Discrete Systems, John Wiley, New York, 1978, p. 363.
[9] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, 1957, p. 12.
2. Department of Mathematics, Faculty of Sciences and Letters, ?stanbul Technical University, ?stanbul 34469, Turkey
3. Department of Mathematics, Faculty of Sciences and Arts, Nam?k Kemal University, Tekirda? 59030, Turkey
Received: September 26, 2011 / Accepted: October 11, 2011 / Published: January 25, 2012.
Abstract: A process represented by nonlinear multi-parametric binary dynamic system is investigated in this work. This process is characterized by the pseudo Boolean objective functional. Since the transfer functions on the process are Boolean functions, the optimal control problem related to the process can be solved by relating between the transfer functions and the objective functional. An analogue of Bellman function for the optimal control problem mentioned is defined and consequently suitable Bellman equation is constructed.
Key words: Bellman equation, bellman function, galois field, shift operator, nonlinear multi-parametric binary dynamic system.
1. Introduction
The utility of nonlinear multi-parametric binary dynamic system (NMBDS) in potential applications in connection with sequential machines, cellular automata and coding theory, the command of technical processes with the help of computers, modeling of the moving objects and simulation of the operation for their control devices, designs of contemporary evaluation systems and other modern engineering problems makes it more actual for research. In this context, an optimal process represented by NMBDS is investigated and Bellman equation interested is derived in this work. The work is organized as follows: Section 2 introduces the process represented by NMBDS. The optimal control problem is presented in section 3. Bellman equation interested is constructed in section 4. Finally, conclusions are given in section 5.
2. NMBDS
Generally, this dynamic system is defined as follows[1-2]:
3. Optimal Control Problem
Now, we can state the original problem represented by NMBDS as follows: In order for a given NMBDS to go from the known initial state ?? to any desired states ??????, to which we expect to access in Lsteps, a control???????[6] must exist such that the functional in (4) has a minimal value:
Since the transfer functions are Boolean and the objective functional which characterizes the process is pseudo Boolean [3, 5], the pseudo Boolean expressions of the transfer functions have been obtained by the operations given in Ref. [7]. After this step, the problem can be stated as follows:
4. Construction of Bellman Equation
For every fixed σandχ, let a function be corresponded to the optimal value of pseudo Boolean functional in the problem (13)-(16). We say that this function is the piecewise analogue of Bellman function[8-9] in the problem (9)-(12):
The initial condition for Bellman equation is given on the right-upper region of ?? and directly determined with the help of the following equality
The condition implying the existence of the unique solution of the system of Eq. (13) is given by Ref. [2],
5. Conclusions
The suitable Bellman equation for piecewise function corresponding to optimal value of the functional in optimal control problem is constructed. It is demonstrated that this piecewise function is the solution to Bellman equation.
References
[1] F. Scheid, Schaum’s Outline of Theory and Problems of Numerical Analysis, McGraw-Hill, Usa, 1988, p. 184.
[2] I. V. Gaishun, Completely Solvable Multidimensional Differential Equations, Nauka and Tekhnika, Minsk, 1983, p. 231.
[3] J.A. Anderson, Discrete Mathematics with Combinatorics, Prentice Hall, New Jersey, 2004, p. 45.
[4] B. Musayev, M. Alp, Functional Analysis, Balc? Publications, Kütahya, Turkey, 2000, p. 7.
[5] S.V. Yablonsky, Introduction to Discrete Mathematics, Mir Publishers, Moscow, 1989, p. 9.
[6] Y.H. Hac?, K. Ozen, Terminal control problem for processes represented by nonlinear multi parameter binary dynamic system, Control and Cybernetics 38 (2009) 625-633.
[7] Y. Hac?, Pseudo Boolean expressions of multi-parametric Boolean vector transfer functions, in: Proceedings of II. Turkish World Mathematics Symposium, Sakarya, Turkey, 2007, p. 151.
[8] V.G. Boltyanskii, Optimal Control of Discrete Systems, John Wiley, New York, 1978, p. 363.
[9] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, 1957, p. 12.