Abstract
We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in . In this article, we discuss the relaxation of such problem.
Highlights
The aim of this article is to analyze the following optimal design problem (OCP), which can be regarded as an optimal control problem, for quasi-linear partial differential equation (PDE) with mixed boundary conditions
We assume that the boundary ∂Ω is Lipschitzian so that the unit outward normal ν =ν ( x) is well-defined for a.e. x ∈ ∂Ω, where the abbreviation ‘a.e.’ should be interpreted here with respect to the ( N −1) -dimensional Hausdorff measure
We need to make sure that minimization problem (19) is meaningful, i.e. there exists at least one pair (u, y) such that (u, y) satisfying the control and state constraints (16)-(18), I (u, y) < +∞, and (u, y) would be a physically relevant solution to the boundary value problem (15)-(16)
Summary
The aim of this article is to analyze the following optimal design problem (OCP), which can be regarded as an optimal control problem, for quasi-linear partial differential equation (PDE) with mixed boundary conditions. Following the standard multiplier rule, which gives a necessary optimality condition for local solutions to state constrained OCPs, the constraint qualifications such as the Slater condition or the Robinson condition should be applied in this case. These conditions cannot be verified for cones such as. We show that this problem admits at least one solution if and only if the corresponding set of feasible solutions is nonempty. In contrast to the Henig relaxation approach, the penalized optimal control problem for indicated variational inequality has a non-empty feasible set and this problem is always solvable. It is unknown whether the entire set of the optimal solutions can be attained in such way
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