Abstract
We consider a damped Kirchhoff-type equation with Dirichlet boundary conditions. The objective is to show the Fréchet differentiability of a nonlinear solution map from a bilinear control input to the solution of a Kirchhoff-type equation. We use this result to formulate the minimax optimal control problem. We show the existence of optimal pairs and find their necessary optimality conditions.
Highlights
Let Ω be an open bounded set of Rn(n ≤ 3) with a smooth boundary Γ
We show the existence of optimal pairs and find their necessary optimality conditions
We show the Frechet differentiability of the solution map of (1): U → y from the bilinear control input variables to the solutions of (1)
Summary
Let Ω be an open bounded set of Rn(n ≤ 3) with a smooth boundary Γ. Hwang and Nakagiri [8] set up optimal control problems developed by Lions [9] with (1) using distributed forcing controls They proved the Gateaux differentiability of the quasilinear solution map from the control variable to the solution and applied the result to derive the necessary optimality conditions for optimal control in some observation cases. We use a firstorder Volterra integrodifferential equation as a proper adjoint equation in the velocity’s observation case, which is another novelty of this paper
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