Optimal Control in Applied Mathematics at Mustansiriyah University
Explore an optimal control problem in applied mathematics at Mustansiriyah University, with state variables prescribed at a fixed terminal time. Assist Prof. Dr. Radhi Ali Zaboon delves into necessary conditions for optimality, boundary conditions, and differential equations to be solved for the control. Discover more about this fascinating topic today!
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Department of Mathematics 2nd Course, 4th class, Applied Mathematics, 2020. Place: Mustansiriyah University, Bagdad, IRAQ. An optimal Control Problem Necessary of Function the State Variables Prescribed at a Fixed Terminal Time By Dr. Radhi A. Zaboon Assist. Prof. Department of Mathematics College of Science, Al_Mustansiriyah University Baghdad, IRAQ. E_Mail: r.zaboon@uomustansiriyah.edu.iq radhizaboon@gmail.com Radhi_maths@yahoo.com Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 1
Where ? is a q-vector ? ? 1 ?? ? = 0, ? ? ?? ? 0 ?? ?? ? ?????? As discussed earlier , the necessary conditions for optimality Are as follows: n-boundary conditions Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com n-boundary conditions 6/3/2025 2
Example: Maximum radius Orbit Transfer in a given Time Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 3
Where From Necessary conditions of optimality & let ? = ?,?,??,? = (??,??,??)? and set ? = ? ?,?,? , where ? ?2 ? ?? ? ? ?? ?+ ?? ? ? ? = 0, ???? ? = ? + ??? ?,?,? = ??? ?,?,? . ??? ?(??) (The objective function) ? ?2+ ????? and the control ? ? ? ?,?,? = ????? Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 4
? = 0 & ? ? = ? ? ?2 ? ?? ? ? ? ? ?2+ ????? ?? ????? ?? and the control ? ? with = ? ? ?+ ? ? Six differential equations are to be solved subject to the six boundary conditions with choice ?1 & ?2 available to satisfy the additional two boundary conditions. The Control ?(?) is determined in terms of ?? & ??. Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 5
Continuous systems; functions of the state variables specified at an unspecified terminal time, including minimum-time problems. ?? ?? ? ?????? The differential is taking into account differential changes in terminal time ?? as: + Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 6
???? + ??? The figure represents the Relationship between ?? ??,? ? ?? ??? ???. ?? ?? ? ? ????????? ?? ? , Means For time held fixed. ?? is standing for differential in ?. ????? ?dt=?????? ?? ?????? ?? ?? =?????? ?? ?? = ??? ?? ? ?? ????dt ??????????? ?? ????? ?? ?? ????dt ?? ????dt ?=?? ?? ?? ????dt ??? (?? ? ? ? )?=?? ?? ??? ?? ? ?=?? = ?? ????dt ??? = ?(?)?=?? ?? Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 7
???? ????? ?dt= ?? ????dt ??? ?? ? ??? ?=?? ?(?)?=?? ?? ?? + Choose the function ? ? to make the coefficients of ?? ? ,?? ? ??? ??? vanish (if ?? is not prescribed) As a result of this choice of the function ? ? See next page Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 8
???? ?????? + (? ???? ? ? ? ?? ) ?? = ?=?? ? ?=????? ?? ???? ?????? + ? ????(??) ( ? ? + = ?? = ?? = ?? ?)?=????? ???? ?????? + ? ????(??) (? + ???? ?????? + ? ????(??) ?(??)??? ?)?=????? For stationary point If a component ???? is not specified, we have ????=0 For minimum time , ?? ?? , we may let ?(? ??,??)=0 and ? = 1 And the ? ????????? ? must be determined to satisfy the terminal constraints Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 9
2n-differential equations The optimality conditions Boundary conditions The optimality conditions determine the ? ?????? ? ? .The 2? differential Equations and the choice of the ? + 1 parameters ? & ? are determined by the 2? + 1 + ? boundary conditions. This boundary value problem is , in general not very easy to solve. Assist. Prof. Dr. Radhi Ali Zaboon, E_Mail: radhi_maths@yahoo.com 6/3/2025 10
