Existence, Uniqueness and Convergence of Simultaneous Distributed-Boundary Optimal Control Problems

We consider a steady-state heat conduction problem $P$ for the Poisson equation with mixed boundary conditions in a bounded multidimensional domain $\Omega$. We also consider a family of problems $P_{\alpha}$ for the same Poisson equation with mixed boundary conditions being $\alpha>0$ the heat transfer coefficient defined on a portion $\Gamma_{1}$ of the boundary. We formulate simultaneous \emph{distributed and Neumann boundary} optimal control problems on the internal energy $g$ within $\Omega$ and the heat flux $q$, defined on the complementary portion $\Gamma_{2}$ of the boundary of $\Omega$ for quadratic cost functional. Here the control variable is the vector $(g,q)$. We prove existence and uniqueness of the optimal control $(\overline{\overline{g}},\overline{\overline{q}})$ for the system state of $P$, and $(\overline{\overline{g}}_{\alpha},\overline{\overline{q}}_{\alpha})$ for the system state of $P_{\alpha}$, for each $\alpha>0$, 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 $P_{\alpha}$ to the corresponding vectorial optimal control, system and adjoint states governed by the problem $P$, when the parameter $\alpha$ 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 $\overline{q}$) and fixed $q$ (with distributed optimal control $\overline{g}$), respectively, for both cases $\alpha>0$ and $\alpha=\infty$.Comment: 14 page

Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems

We consider a steady-state heat conduction problem in a multidimensional bounded domainfor the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion 1 of the boundary and a constant heat flux q in the remaining portion2 of the boundary.Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary 1 with heat transfer coefficient α and external temperature b. We obtain explicitly, for a rectangular domain in R2, an annulus in R2 and a spherical shell in R3, the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on 1 converge, when α → ∞, to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on 1. Also, we analyze the order of convergence in each case, which turns out to be 1/α being new for these kind of elliptic optimal control problems.Fil: Bollati, Julieta. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Gariboldi, Claudia Maricel. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin

Convergence of simultaneous distributed-boundary parabolic optimal control problems

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 α 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.Fil: Tarzia, Domingo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral; ArgentinaFil: Bollo, Carolina María. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Fisicoquímicas y Naturales. Departamento de Matemática; ArgentinaFil: Gariboldi, Claudia Maricel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Fisicoquímicas y Naturales. Departamento de Matemática; Argentin

Numerical analysis of a family of simultaneous distributed-boundary mixed elliptic optimal control problems and their asymptotic behaviour through a commutative diagram and error estimates

In this paper, we consider a family of simultaneous distributed-boundary optimal control problems ($P_{\alpha}$) on the internal energy and the heat flux for a system governed by a mixed elliptic variational equality with a parameter $\alpha >0$ and a simultaneous distributed-boundary optimal control problem ($P$) governed also by an elliptic variational equality with a Dirichlet boundary condition on the same portion of the boundary. We formulate discrete approximations $\left(P_{h \alpha}\right)$ and $\left(P_h\right)$ of the problems $\left(P_\alpha\right)$ and $(P)$ respectively, for each $h>0$ and for each $\alpha>0$, through the finite element method with Lagrange's triangles of type 1 with parameter $h$ (the longest side of the triangles). The goal of this paper is to study the convergence of this family of discrete simultaneous distributed-boundary mixed elliptic optimal control problems $\left(P_{h \alpha}\right)$ when the parameters $\alpha$ goes to infinity and the parameter $h$ goes to zero simultaneously. We prove the convergence of the problems $\left(P_{h \alpha}\right)$ to the problem $\left(P_h\right)$ when $\alpha \rightarrow +\infty$, for each $h>0$. We study the convergence of the problems $\left(P_{h \alpha}\right)$ and $\left(P_h\right)$, for each $\alpha >0$, when $h \rightarrow 0^+$ obtaining a commutative diagram which relates the continuous and discrete optimal control problems $\left(P_{h \alpha}\right),\left(P_\alpha\right),\left(P_h\right)$ and $(P)$ by taking the limits $h \rightarrow 0^+$ and $\alpha \rightarrow +\infty$ respectively. We also study the double convergence of $\left(P_{h \alpha}\right)$ to $(P)$ when $(h, \alpha) \rightarrow(0^+,+\infty)$ which represents the diagonal convergence in the above commutative diagram.Comment: This paper has been published online in Nonlinear Analysis: Real World Applications. arXiv admin note: text overlap with arXiv:1512.0383

A Convergence Criterion for Elliptic Variational Inequalities

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 $\{u_n\}\subset X$ which guarantee the convergence $u_n\to u$ 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.Comment: 26 pages. arXiv admin note: text overlap with arXiv:2005.1178

Existence, comparison, and convergence results for a class of elliptic hemivariational inequalities

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.Comment: 22 page

Activation of Enteroendocrine Cells via TLRs Induces Hormone, Chemokine, and Defensin Secretion

Abstract Enteroendocrine cells are known primarily for their production of hormones that affect digestion, but they might also be implicated in sensing and neutralizing or expelling pathogens. We evaluate the expression of TLRs and the response to specific agonists in terms of cytokines, defensins, and hormones in enteroendocrine cells. The mouse enteroendocrine cell line STC-1 and C57BL/6 mice are used for in vitro and in vivo studies, respectively. The presence of TLR4, 5, and 9 is investigated by RT-PCR, Western blot, and immunofluorescence analyses. Activation of these receptors is studied evaluating keratinocyte-derived chemokine, defensins, and cholecystokinin production in response to their specific agonists. In this study, we show that the intestinal enteroendocrine cell line STC-1 expresses TLR4, 5, and 9 and releases cholecystokinin upon stimulation with the respective receptor agonists LPS, flagellin, and CpG-containing oligodeoxynucleotides. Release of keratinocyte-derived chemokine and β-defensin 2 was also observed after stimulation of STC-1 cells with the three TLR agonists, but not with fatty acids. Consistent with these in vitro data, mice showed increased serum cholecystokinin levels after oral challenge with LPS, flagellin, or CpG oligodeoxynucleotides. In addition to their response to food stimuli, enteroendocrine cells sense the presence of bacterial Ags through TLRs and are involved in neutralizing intestinal bacteria by releasing chemokines and defensins, and maybe in removing them by releasing hormones such as cholecystokinin, which induces contraction of the muscular tunica, favoring the emptying of the distal small intestine

Simultaneous optimal controls for non-stationary Stokes systems

International audienceThis paper deal with optimal control problems for a non-stationary Stokes system. We study a simultaneous distributed-boundary optimal control problem with distributed observation. We prove the existence and uniqueness of a simultaneous optimal control and we give the first order optimality condition for this problem. We also consider a distributed optimal control problem and a boundary optimal control problem and we obtain estimations between the simultaneous optimal control and the optimal controls of these last ones. Finally, some regularity results are presented

Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

International audienceWe consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by $\alpha$ the friction coefficient and we analyze the asymptotic behavior of such a problem as $\alpha\to\infty$. More precisely, we prove that if we take an optimal control for each $\alpha$, 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