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Optimal control of quasilinear elliptic equations with non differentiable coefficients at the origin. (English) Zbl 0749.49006

The authors study some optimal control problems involving the differential operator: \(Ay=-\text{div}(u(x,|\nabla y|)\nabla y)+v(x,y)\), with \(u: \Omega\times(0,\infty)\to(0,\infty)\) and \(v: \Omega\times R\to R\), where \(\Omega\) is a bounded open subset of \(R^ N\) with Lipschitz continuous boundary. In this paper, the non- differentiability of \(u(x,\cdot)\) at 0 is allowed, which causes the non- differentiability of state with respect to the control. The existence of a solution is proved. The optimality conditions are obtained by considering a perturbation of the coefficients of the differential operator \(A\).

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
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