Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 1: Theory |
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| Authors: | J M Sloss I S Sadek J C Bruch Jr S Adali |
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| Institution: | (1) Department of Mathematics, University of California, Santa Barbara, California;(2) Department of Mathematical Sciences, University of North Carolina, Wilmington, North Carolina;(3) Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California;(4) Department of Mechanical Engineering, University of Natal, Durban, South Africa |
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| Abstract: | A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression. |
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| Keywords: | Maximum principle distributed-parameter systems optimal control structural control hyperbolic partial differential equations |
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