Abstract
In this paper, we study quadratic cost optimal control problems governed by a von Karman system with long memory. We prove the existence of an optimal control for the cost. Then, by proving the strong Gâteaux differentiability of nonlinear solution mapping we establish necessary optimality condition for the optimal control corresponding to the quadratic cost. Further, we study the time local uniqueness of the optimal controls for distributive observation.
Highlights
We consider a von Kármán plate model with internal damping and long memory
In order to apply the variational approach due to Lions [ ] to our problem, we propose the quadratic cost functional J as studied in Lions [ ], which is to be minimized within Uad, an admissible set of control variables in U
Following the idea in [ ], we show the strict convexity of the quadratic cost J for distributive observation in local time interval by making use of the second-order Gâteaux differentiability of the nonlinear solution mapping u → y(u)
Summary
We consider a von Kármán plate model with internal damping and long memory. In the context of control theory, early results for the von Kármán plate can be found in [ ], which gives the derivation of the model and asymptotic energy estimates for the system.In this paper, our system may be described as follows: Let be an open bounded domain in R with a sufficiently smooth boundary ∂. Based on this result, we study the following optimal control problem: Minimize J(u) Following the idea in [ ], we show the strict convexity of the quadratic cost J for distributive observation in local time interval by making use of the second-order Gâteaux differentiability of the nonlinear solution mapping u → y(u).
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