DCDS-B
On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems
Olha P. Kupenko Rosanna Manzo
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1363-1393 doi: 10.3934/dcdsb.2018155

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 $u(x)∈ L^∞(Ω)\cap BV(Ω)$ 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 $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$. We show that, in spite of unboundedness of the non-linear 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.

keywords: Generalized p-Laplace equations control in coefficients variational convergence
DCDS-B
On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity
Peter I. Kogut Olha P. Kupenko
Discrete & Continuous Dynamical Systems - B 2019, 24(3): 1273-1295 doi: 10.3934/dcdsb.2019016

We study an optimal control problem for one class of non-linear elliptic equations with $p$-Laplace operator and $L^1$-nonlinearity. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any given control. After defining a suitable functional class in which we look for solutions, we reformulate the original problem and prove the existence of optimal pairs. In order to ensure the validity of such reformulation, we provide its substantiation using a special family of fictitious optimal control problems. The idea to involve the fictitious optimization problems was mainly inspired by the brilliant book of V.S. Mel'nik and V.I. Ivanenko "Variational Methods in Control Problems for the Systems with Distributed Parameters", Kyiv, 1998.

keywords: Existence result optimal control $p$-Laplace operator elliptic equation fictitious control
NHM
On boundary optimal control problem for an arterial system: First-order optimality conditions
Ciro D'Apice Olha P. Kupenko Rosanna Manzo
Networks & Heterogeneous Media 2018, 13(4): 585-607 doi: 10.3934/nhm.2018027

We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that the control object is described by an initial-boundary value problem for $ 1D $ system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principle part and the Robin-type of boundary conditions. The main question we study in this part of the paper is about the existence of optimal solutions and first-order optimality conditions.

keywords: Boussinesq-type system existence result first-order optimality conditions optimal solutions

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