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2012 - 11th Annual NC State Summer Undergraduate Research Symposium
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Session Time :
8/1/12 3:00 PM - 8/1/12 4:14 PM
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Content Area : OIA-SRE, Office of International Affairs - Summer Research Experience
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Lead Student Presenters : Yu Xia
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Abstract Title : An effective numerical method based on PMP and HJB equation
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Abstract :
The aim of this research is to design an effective numerical scheme for optimal control problems. Depending on the existence of optimal control, a currently popular scheme to solve optimal control problems is the shooting method, which is based on Pontryagin’s Maximum Principle. Pontraygin’s Maximum Principle is part of the necessary conditions that an optimal control along with corresponding optimal trajectory of a control problem must satisfy. In Pontryagin’s maximum principle the adjoint variable, which is part of the necessary conditions, is not determined at the initial point of the time horizon. Instead it is known at the end of the time horizon. Thus, the optimal state and adjoint variable constitute a boundary value problem of coupled differential equations. The Hamilton Jacobi Bellman Theory provides a characterization of the optimal value of the objective functional based at points of the time and space coordinate system. This optimal value is called the value function. It satisfies a first order partial differential equation. It is, in general, difficult to solve this partial differential equation. However, a rough approximation can be obtained by numerical computation. The approximate value function can then be used to make a useful guess of the initial value of the adjoint problem improving the effectiveness of the shooting method. I will give two examples to illustrate the approach. The first example is a general control problem and the second comes from a problem in marketing.
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Mentor and/or Co-Author : Negash G. Medhin
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