Abstract
In this paper we study the existence and relaxation theory for distributed parameter systems driven by nonlinear elliptic differential equations. The existence theory is developed for systems with a priori feedback (closed loop systems), while the main results of the relaxation results are proved for open loop systems. We start with some existence theorems for nonlinear elliptic inclusions which are then used to establish the existence of admissible states for the optimal control problem. Then with the help of a convexity-type condition, we formulate and prove the main existence theorem for the problem. The proof employs the so-called reduction technique. Then we formulate three different versions of the relaxed problem, we show that they are equivalent and finally we show that they have the same value as the original problem and their set of states captures the asymptotic behavior of the sequences of original states.
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Kourogenis, N.C., Papageorgiou, N.S. Optimization and relaxation of nonlinear elliptic control systems. Japan J. Indust. Appl. Math. 17, 453–479 (2000). https://doi.org/10.1007/BF03167377
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DOI: https://doi.org/10.1007/BF03167377