Abstract
In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality.
Highlights
We consider an optimal control problem associated with the following elliptic semilinear equation
Where A is an elliptic operator, b : Ω −→ Rn is a given function, f : Ω × R −→ R is nondecreasing monotone in the second variable, u ∈ L2(Ω), Ω is a domain in Rn, n = 2 or 3, and Γ is the boundary of Ω
Due to the convection term induced by b, the linear part of the above operator is nonmonote
Summary
We consider an optimal control problem associated with the following elliptic semilinear equation. A thorough study is needed to prove existence and uniqueness of a solution of the equation (1.1) for every u. Only a few references treat the topic of existence and uniqueness of solution for linear elliptic equations with convection term such that div b = 0 and b is not small. The equation considered in this paper does not fit in the problems studied in the mentioned references. Associated with the state equation (1.1) we consider the following control problem:. A precise analysis of the state equation allows us to prove the existence of a solution for (P) as well as to get the first and second order optimality conditions. Based on the results established in this paper, the numerical analysis for (P) will be carried out in a forthcoming paper
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