On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

Abstract

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain Ω. We take the coefficient u(x) ∈ L ∞ (Ω) ∩ BV (Ω) in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix A skew ∈ L q (Ω; S N skew). We show that, in spite of unboundedness of the nonlinear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices A skew. We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

A Fluid Dynamic Approach To Model And Optimize Energy Flows In Networked Systems

“…states. Indeed, the superiority of the LWR model is also due to results of existence and uniqueness of solutions for large networks, guaranteeing a solid analytical theory for numerical approximations and optimization problems (similar drawbacks are also described for different types of PDEs in [20] and [21] and for energy issues in [22]). For instance, in [23], efficient numerical algorithms are described to treat complex networks in acceptable computational times.…”

An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

“…Several results for optimal control problems related to elliptic PDE's with unbounded coefficients have been also obtained in [26,21,22,27,28,9,10] and the recent papers [20], [30] with the reference therein).…”

On optimal <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-control in coefficients for quasi-linear Dirichlet boundary value problems with <inline-formula><tex-math id="M2">\begin{document}$ BMO $\end{document}</tex-math></inline-formula>-anisotropic <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian

“…We note that these assumptions on the class of matrices are essentially weaker than they usually are in the literature (see, for instance, [9,10,13,21,22]). In fact, we deal with a Dirichlet boundary value problem (BVP) for degenerate anisotropic elliptic equation with unbounded coefficients in its principal part and with L 1bounded control in the coefficient of the low-order term.…”