Abstract
We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted $p$-Laplace problem. The coefficient of the $p$-Laplacian, the weight $u$, we take as a control in $BV(\Omega)\cap L^\infty(\Omega)$. In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the $p$-Laplacian, we use a regularization, sometimes referred to as the $\varepsilon$-$p$-Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted $\varepsilon$-$p$-Laplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by $k$. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each $(\varepsilon,k)$-level as the parameters tend to zero and infinity, respectively.
Highlights
Control in the coefficients of elliptic problems has a long history of its own, starting with the work of Murat [10, 11] and Tartar [14]
The constrained optimal control problem (OCP) in the coefficients of the leading order differential expressions was first discussed in detail by Casas [2] in the case of the classical Laplace equation, where the scalar coefficient u in the div(u∇·) formulation was taken as control satisfying box constraints with a strictly positive lower and some upper bound together with a slope constraint
While the ε-p-Laplacian regularizes the degeneracy as the gradients tend to zero, the term u|∇y|p−2, viewed again as a coefficient for the otherwise linear problem, may grow large. We introduce yet another regularization that leads to a sequence of monotone and bounded approximation Fk(|∇y|2) of |∇y|2
Summary
Control in the coefficients of elliptic problems has a long history of its own, starting with the work of Murat [10, 11] and Tartar [14]. Since the set Ξ is nonempty and the cost functional is bounded from below on Ξ, it follows that there exists a minimizing sequence {(uk, yk)}k∈N ⊂ Ξ to problem (P), i.e., inf I(u, y) = lim. As follows from Theorem 3.5, this fact is not an obstacle to proving existence of optimal controls in the coefficients, but it causes certain difficulties when deriving the optimality conditions for the considered problem To overcome this difficulty, we introduce the following family of approximating control problems (see, for comparison, the approach of Casas and Fernandez [3] for quasilinear elliptic equations with a distributed control in the right-hand side):. It is clear that the effect of such perturbations of Δp(u, y) is its regularization around critical points where |∇y(x)| lar, if y ∈ W01,p(Ω) and Ωk(y) := x vanishes or becom√es unbounded. ∈ Ω : |∇y(x)| > k2 + 1 ,
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