Procedure for asymptotic state and parameter estimation of nonlinear distributed parameter bioreactors

Due to measurement problem in biological variables, such as the biomass concentration and its specific growth rate, the on-line measurement of these variables is not ensured and this presents a great difficulty, especially when we consider the control problem. This paper presents and analyses an approach for estimating these biological variables through an example of a nonlinear distributed parameter bioreactor. An orthogonal collocation method is applied in the distributed system to obtain a system of ordinary differential equations. A simulation study shows the feasibility and the robustness of the estimator used for this nonlinear process.     

Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

On an evolution equation with acoustic boundary conditions

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota‐Goldstein work in the sense of considering a weaker internal damping and one more quadratic nonlinearity in the elastic string equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive servomechanism design for boundary control systems

In this paper we consider the design of adaptive servomechanismfor boundary control systems in the case of the input and outputoperators being collocated. An adaptive servomechanism is constructedby the concept of high-gain output feedback and the estimationmechanism of the unknown parameters for the bounded disturbances.Moreover, a controller is designed to be inverse optimal. Itminimizes a meaningful cost functional that incorporates integralpenalty on the tracking error state, the control and the parametererror.     

Nontrivial solutions for asymptotically linear hamiltonian systems with Lagrangian boundary conditions

In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.     

On the properties of solutions to the Goursat-Darboux problem with boundary and distributed controls

We consider a control system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls. The controls are subject to the constraints given as multivalued mappings with closed, possibly nonconvex, values depending on the phase variable. Alongside the initial constraints, we consider the convexified constraints and the constraints whose values are the extreme points of the convexified constraints. We study the questions of existence of solutions and establish connections between the solutions under various constraints.     

Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions

This paper concerns with blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space. We establish the rate estimates for blow-up solutions and prove that the blow-up set is under proper conditions on initial data. Furthermore, for N=1, more complete conclusions about such two topics are given.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.     

Matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was previously known only in the second-order case and fourth-order case with separated boundary conditions.     

On the three-dimensional Wigner-Poisson problem with inflow boundary conditions

We study the Wigner-Poisson problem in a bounded spatial domain, with non-homogeneous and time-dependent “inflow” boundary conditions. This system is a quantum model of charge transport in a semiconductor device coupled with reservoirs, in presence of a self-consistent potential and of an external one. We state a local-in-time well-posedness result for the problem. The main difficulty is proving in the three-dimensional case that the non-linear potential term is a Lipschitz perturbation of the “affine” streaming operator, in an appropriately weighted L²-space.     

Small sets for Neumann boundary conditions

For an open set D ? ?ⁿ and a relatively closed subset E ? D of Lebesgue measure zero, we investigate conditions for the property that Brownian motion with reflexion at the boundary on D and D \ E are the same. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions. By constructing the Poincare operator, we obtain the existence of -periodic weak solutions under some reasonable assumptions.     

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in

In this paper, we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in by a solution to a Cauchy problem of the Laplace equation. Assuming that the unknown part is a Lipschitz continuous surface, we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part. The keys are the complex extension and estimates for a harmonic measure.

     

On the Laplace equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the Laplace equation in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem