Search

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence $ \{u_n\}\subset X $ { u n } ⊂ X which guarantee the convergence $ u_n\to u $ u n → u in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study boundary optimal control problems on the heat flux and simultaneous distributed-boundary optimal control problems on the internal energy and the heat flux for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system was originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the boundary optimal control problem and simultaneous distributed-boundary optimal control problems. We show an asymptotic behavior result for the optimal controls and the system states for both optimal control problems, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states. [ABSTRACT FROM AUTHOR]

We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain with regular boundary and a family of problems Sα with also mixed boundary conditions in, where α > 0 is the heat transfer coefficient on the portion of the boundary T1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion T2 of the boundary of. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα. We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity. [ABSTRACT FROM AUTHOR]

We consider a heat conduction problem S with mixed boundary conditions in a n-dimensional domain Ω with regular boundary Γ and a family of problems Sα, where the parameter α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate simultaneous distributed-boundary optimal control problems on the internal energy g and the heat flux q on the complementary portion of the boundary Γ2. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author. [ABSTRACT FROM AUTHOR]

We consider a steady-state heat conduction problem P for the Poisson equation with mixed boundary conditions in a bounded multidimensional domain Ω. We also consider a family of problems Pa for the same Poisson equation with mixed boundary conditions, α > 0 being the heat transfer coefficient defined on a portion ⌈1 of the boundary. We formulate simultaneous distributed and Neumann boundary optimal control problems on the internal energy g within Ω and the heat flux q, defined on the complementary portion ⌈2 of the boundary of Ω for quadratic cost functional. Here, the control variable is the vector (g,q). We prove existence and uniqueness of the optimal control (...) for the system state of P, and (...) for the system state of Pa, for each α > 0, and we give the corresponding optimality conditions. We prove strong convergence, in suitable Sobolev spaces, of the vectorial optimal controls, system and adjoint states governed by the problems Pa to the corresponding vectorial optimal control, system and adjoint states governed by the problem P, when the parameter a goes to infinity. We also obtain estimations between the solutions of these vectorial optimal control problems and the solution of two scalar optimal control problems characterized by fixed g (with boundary optimal control ...) and fixed q (with distributed optimal control ...), respectively, for cases both of α > 0 and α = ∞. [ABSTRACT FROM AUTHOR]

We consider a steady-state heat conduction problem Pα with mixed boundary conditions for the Poisson equation depending on a positive parameter α, which represents the heat transfer coefficient on a portion Γ[SUB1] of the boundary of given bounded domain in R[SUPn]. We formulate distributed optimal control problems over the internal energy g for each α. We prove that the optimal control g[SUBop[SUBα]] and its corresponding system u[SUBg[SUBop[SUBα]α]] and adjoint p[SUBg[SUBop[SUBα]α]] states for each a are strongly convergent to g[SUBop],u[SUBg[SUBop]] and p[SUBg[SUBop]], respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control, and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation with a different boundary condition on the portion Γ[SUB1].We use the fixed point and elliptic variational inequality theories. [ABSTRACT FROM AUTHOR]