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In this paper an approximation theory is provided for the solutions of infinite dimensional Algebraic Riccati Equations, which in particular includes convergence of the approximating Riccati operators as well as convergence of the approximating gain operators. The main features which distinguish this paper from other work existing in the literature of Riccati approximation theory are: (i) the original free system only generates a strongly continuous semigroup; (ii) the input (control) operator is generally unbounded; and (iii) no “smoothing” hypothesis on the observation operator is assumed. The abstract theory is illustrated by several examples arising in boundary control problems for wave and plate equations.

A new type of meshless method is proposed in this paper to solve regularized Laplacian boundary value problems of the form L u : = r 2 u − Δ u ≡ 0 . In this method, solving the Laplacian boundary v...

In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary...

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. ...

This paper deals with the problem − Δ p u + α(x)|u| p−2 u = β(x)f(|u|) in Ω, subjected to the zero Neumann boundary condition, where p > 1, Ω ⊂ ℝ N is bounded with smooth boundary, α, β ∊ L ∞(Ω), essinfΩβ > 0, and f:[0,+ ∞) → ℝ is a not necessarily continuous nonlinearity that oscillates either at the origin or at the infinity. By using nonsmooth variational methods, we establish in both cases the existence of infinitely many distinct non-negative solutions of the Neumann problem. In our framework, α:Ω → ℝ may be a sign-changing or even a nonpositive potential, which is not permitted usually in earlier works.

This paper deals with the existence and necessary optimality conditions for an optimal control problem governed by a phase-field transition system. The one-point boundary (time variable) state condition is considered. A numerical algorithm of gradient type and numerical implementation are reported, too.

This paper is concerned with the existence and uniform decay for the solutions to the multivalued hyperbolic differential inclusion with memory condition at the boundary.

This paper is concerned with the analysis of the discontinuous Galerkin finite element method applied to the space semidiscretization of nonlinear nonstationary convection-diffusion problems. The attention is paid to the derivation of the method and the investigation of error estimates for schemes with a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The theoretical results are illustrated by numerical experiments.

This paper describes some properties and applications of Steklov eigenproblems for prototypical second-order elliptic operators on bounded regions in ℝ n . Results are described for Schroedinger and weighted harmonic equations. A variational description of the least eigenvalue leads to optimal L 2-trace inequalities. It is shown that the eigenfunctions provide complete orthonormal bases of certain closed subspaces of H 1(Ω) and also of L 2(∂Ω, dσ). This allows the description, and representation, of solution operators for homogeneous elliptic equations subject to inhomogeneous Dirichlet, Neumann or Robin boundary data. They are also used to describe Robin to Dirichlet and Neumann to Dirichlet operators for these equations, and to describe the spectrum of these operators. The allowable regions are quite general; in particular classes of bounded regions with a finite number of disjoint Lipschitz components for the boundary are allowed.

In this paper we introduce a boundary coercive condition on the regularizing function of the proximal point method in Banach spaces, which has a penalization effect and is useful for solving certain variational inequality problems. As its finite dimensional counterpart (proximal point method with Bregman distances) it avoids explicit consideration of the constrains, because the feasibility is taken care of by the regularizing function, whose derivative diverges on the boundary of the feasible set. The algorithm is therefore an interior point one. We show that this effect is guaranteed even for an inexact version of the method, namely the hybrid extragradient proximal point method proposed by Solodov and Svaiter for finite dimensional spaces. We give a full convergence analysis, and examples of regularizing functions satisfying the required conditions, for the cases of the feasible set being a closed ball or a polyhedron.

In this paper we consider a groundwater flow problem in a semi-infinite vertical region bounded on top by a sloping sinusoidal curve dealt with by Shivakumar et al. (Shivakumar, P.N.; Williams, J.J.; Qiang Ye.; Chuanxiang, Ji. An Analysis of Groundwater Flow in an Infinite Region with Sinusoidal Top. Numer. Funct. Anal. and Optimiz. 2000, 21(1&2), 263–271). We first extend the analysis to the more realistic case of a finite region Ω. The problem is described by an elliptic equation for the hydraulic head under given boundary conditions. The obtained results are in good agreement with a classical result (Toth, J. A Theoretical Analysis of Groundwater Flow in Small Drainage Basins. J. Geophys. Res. 1963, 67, 4795–4812). We extend the problem further by allowing Dirichlet-type boundary conditions on the lower boundary. Our first goal is to obtain a semi-analytical solution. The problem is reduced to an infinite system of linear equations by using a suitable Fourier expansion method, transforming low...

The purpose of the paper is to present a unified numerical method for problems consisting of the conventional boundary value problem. Cauchy problem, under-determined problem, and over-determined problem. The method is based on the direct variational approach, which paraphrases the problems into the primary and adjoint boundary value problems that can be tackled by commonly used computer programs for the numerical solution of the Laplace equation.

The method of orienting curves was developed for solving some classes of optimal control problems with state constraints. In the present paper, this method is transferred to the problem of finding the upper bound of a class of functions with fixed end points and restrictions to their value and their derivative. The algorithm presented here delivers constructively the optimal solution as a path consisting of orienting curves, boundary arcs, and final curves. The result is applied to determining the time-optimal motion of a robot following a prescribed trajectory under some restrictions to its velocity and acceleration.

Earlier results for distributed and boundary controls of the viscous Burgers' equation were established by Burns et al. and Byrnes et al. In their results there are technical restrictions on the sizes of the initial data. In this paper we relax these restrictions, as well as treat the Burgers' equation with other nonlinear boundary conditions.

The purpose of this paper is to prove error bounds for the standard Galerkin piecewise linear finite element approximation of the following Model Vortex Problem: Given a bounded, connected domain with boundary ∂Ω e C 2,$gamma; for some γ > 0, and constants p e [0,1) and find u such that The main mathematical interest of P lies in the fact that it is a semilinear elliptic free boundary problem with a non–Lipschitz nonlinearity. The existence of a well-defined maximal branch of solutions of P is shown for d sufficiently small. Using a monotonicity and maximum principle approach, an L ∞ error bound of order , where, δ e (0,1), is proved for a finite element approximation of P involving exact integration of the nonlinear term, provided that the maximal solution branch of the continuous problem is Holder continuous with exponent with respect to d. This condition is then shown to hold for any δ e (0,1), for p e (0,1) for and d sufficiently small. Thus the stated error bound is near-optimal.

In this paper we consider two mathematical models for a multiple beam system (MBS) which is composed of two rigidly and angularly connected Euler-Bernoulli beams. The cantilevered structure is clamped at one end, and has point controls for forces and bending moments imposed at the other end and at the connection between the two beams. The first model incorporates not only transverse deformations of both beams, but also axial compression/extension of the beams. The second model involves only transverse deformations of the beam. By imposing point controls, an unbounded input operator is obtained. A variational formulation of the models is used to show well-posedness. Uniform exponential stabilizability of the second model through boundary feedback is established via energy arguments.

In this paper, we apply the coupling of the boundary integral and finite element methods to study the weak solvability of certain nonmonotone nonlinear exterior boundary value problems. In order to convert the original exterior problem into an equivalent nonlocal boundary value problem on a finite region, we employ two different approaches based on the use of one and two integral equations on the coupling boundary. Existence of a solution for the associated weak formulation, and convergence properties of the corresponding Galerkin approximations are deduced from fundamental results in nonlinear functional analysis. Indeed, the main arguments of our proofs are based on a surjectivity theorem for mappings of type (S) and on the Fredholm alternative for nonlinear A-proper mappings.

This paper is concerned with a class of optimal control problems with state and control constraints which have one state variable but several control variables. The variables can appear linearly or not linearly, and the performance index and the function in the state equation are able to be non-convex in their variables. In order to overcome difficulities in solving such problems that are caused by state constraints, the Method of Orienting Curves is developed. By this method, the optimal trajectory is constructed as a path which consists of parts of orienting curves, boundary arcs, and finally, of the final curve. This method can be applied tho solve a lot of practical problems, for example, optimal control of hydroelectric power plants, Zemerlo's navigation problem with state constraints …. Here, an inventory problems is considered as application example.

This paper is concerned with the numerical solution of optimal control problems for which each optimal control is bang-bang. Especially, the results apply to parabolic boundary control Problems. Starting from a sequence of feasible solutions converging to an optimal control u, a sequence of bang-bang controls converging to u is constructed. Bang-bang approximations of u are desirable for certain numerical reasons. Sequences of arbitrary feasible controls converging to u may be obtained by discretization or by a descent method. Numerical examples are also given.

In this paper a system of partial differential equations is considered that constitutes a one-dimensional model for the dynamics of martensitic phase transitions in Shape Memory alloys. The stability behaviour of the system under distributed heat sources and loads and heat sources at the boundary is investigated. An optimal control problem is formulated which uses loads and heat sources as control variables in order to induce both stress- and temperature-induced phase transitions. The existence of optimal controls is proved.

This paper deals with the time-optimal problem for boundary control of parabolic differential equations. In addition to the usual control constraint a rather general state constraint is admitted. The target set is assumed to have an interior point with respect to the topology induced by the maximum norm. An existence theorem and a maximum principle are derived for this problem. Moreover, for a special case a generalized bang-bang principle is proved.

In this paper, we consider an optimal control problem involving a class of first order hyperbolic systems with boundary controls. A computational algorithm which generates minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated. Moreover, a necessary and sufficient condition for optimality is derived and a result on the existence of optimal controls is obtained.