Maximum principle of a class of nonlinear PDE systems
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In Banach spaces with either L2or L∞norm, a generalized integral version of the maximum principle for a class of nonlinear distributed-parameter systems (DPSs) and a generalized cost functional is obtained, and its pointwise version is proved to be equivalent to this integral form with an additional assumption on controls. The Hamilton-Jacobi equation for the DPSs is also obtained. To illustrate the use of the theory, the maximum principle and the Hamilton-Jacobi equation are applied to the design of optimal open-loop and feedback controls for a two-dimensional, nonlinear, hyperbolic system. The open-loop optimal control law is obtained without introducing model linearization and truncation. In the space with L∞-type norm the optimal feedback control law structure is obtained from the open-loop result. Two approximations to the optimal feedback control are analyzed, and the approximate errors are estimated in terms of the initial state norm.
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