On a generalised<Emphasis Type="Italic">ε</Emphasis>-control problem in Banach space

In this paper a linear 49;-control problem for a general objective function is considered by functional analytic approach. Existence and uniqueness of solutions are shown. The characterization of the solution is given in terms of a hyperplane.     

Estimates for solutions of quasilinear problems with dead cores

A Rayleigh-Faber-Krahn type inequality is used to derive bounds for boundary value problems appearing in reaction-diffusion problems where the reactant is consumed. Interesting quantities are the minimum of the solution and the measure of the set where it vanishes. The proofs are rather elementary and apply to problems possessing solutions in a weak sense.     

Weak compactness in certain star-shift invariant subspaces

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form , where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence in K_B such that the sequence of real measures converges weak-star to some nontrivial singular measure on ? We show that, in order for this to hold, K_B must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that K_B has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

Convergence of Sobolev spaces on varying manifolds

We deal with variational problems on varying manifolds in ℝⁿ. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝⁿ. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μ_h -a.e., being {μ_h} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs (P_{μ h} H(μ_h)), where P_{μ h} are the projectors onto the tangent spaces T_{μ h}. When μ_h belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary.     

On hypersurfaces inR n+1 with integral bounds on curvature

We show that the L ^p norm of the second fundamental form of hypersurfaces in R ⁿ⁺¹ is very coercive in the GMT setting of Gauss graphs currents, when p exceeds the dimension n. A compactness result for immersed hypersurfaces and its application to a variational problem are provided.     

Stochastic optimal control and algorithm of the trajectory of horizontal wells

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.     

Continuous dependence results for non-linear Neumann type boundary value problems

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation.     

Maps leaving functional values of operator products invariant

Let A be a standard operator algebra on a complex Hilbert space H of dimension greater than 2. By invariants of certain functional values of operator products, we characterize some surjective maps on A. Furthermore, several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as, for example, k-numerical radius (k?1), operator norm, Ky Fan k-norm, Schatten p-norm (1?p<∞), and so on.