共查询到20条相似文献,搜索用时 31 毫秒
1.
David R. AdamsSuzanne Lenhart 《Journal of Mathematical Analysis and Applications》2002,268(2):602-614
An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L2(0, T; H2(Ω) ∩ H10(Ω)) with ψt ∈ L2(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established. 相似文献
2.
An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the
control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H
1
0
(Ω) so that the state is close to the desired profile while the H
1
(Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal
pairs are established.
Accepted 11 September 1996 相似文献
3.
We consider an optimal control problem where the state satisfies an obstacle type semilinear variational inequality and the control function is the obstacle. The state is chosen to be close to a desired profile while the obstacle is not too large in H
0
1
(), and H
2-bounded. We prove that an optimal control exists and give necessary optimality conditions, using approximation techniques. 相似文献
4.
《Applied Mathematics Letters》2001,14(6):769-773
In this paper, we consider a numerical enclosure method with guaranteed L∞ error bound for the solutions of obstacle problems. Using the finite-element approximations and the explicit a priori error estimates for obstacle problems, we present an effective verification procedure that automatically generates on a computer a set which includes the exact solution. A particular emphasis is that our method needs no assumption of the existence of the solution of the original obstacle problems, but it follows as the result of computation itself. A numerical example for an obstacle problem is presented. 相似文献
5.
In this paper, we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite elements and the control is discretized by piecewise constant functions. We prove the superconvergence error estimate of h2 in L2-norm between the approximated solution and the interpolation of the exact control variable. Moreover, by postprocessing technique, we find that the projection of the discrete adjoint state is superclose (in order h2) to the exact control variable. 相似文献
6.
We deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle. We assume that the boundary of the obstacle is a non-analytic hypersurface. We show that, when we put Dirichlet boundary conditions, one measurement is enough to reconstruct the obstacle. In the Neumann case, we have results only for n = 2, 3 in general. More precisely, we show that one measurement is enough for n = 2 and we need 3 linearly independent inputs for n = 3. However, in the case for the Helmholtz equation, we only need n ? 1 linearly independent inputs, for any n ≥ 2. Here n is the dimension of the space containing the obstacle. These are justified by investigating the analyticity properties of the zero set of a real analytic function. In addition, we give a reconstruction procedure for each case to recover the shape of obstacle. Although we state the results for the scattering problems, similar results are true for the associated boundary value problems. 相似文献
7.
In this paper, the parabolic quasi-variational inequalities are transformed into a noncoercive elliptic quasi-variational inequalities. A new iterative discrete algorithm is proposed to show the existence and uniqueness, and a simple proof to asymptotic behavior in uniform norm is also given using the theta time scheme combined with a finite element spatial approximation. The proposed approach stands on a discrete L∞-stability property with respect to the right-hand side and obstacle defined as an impulse control problem. 相似文献
8.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(3):281-286
We prove an estimate for the Dirichlet-Neumann operator, and for the H1 local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward 相似文献
9.
Felipe álvarez Jér?me Bolte J. Frédéric Bonnans Francisco J. Silva 《Mathematical Programming》2012,135(1-2):473-507
We consider a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we provide a first order expansion for the penalized states and adjoint states around the state and adjoint state of the original problem. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian except for a set of times with null Lebesgue measure, the functional estimates for the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide several types of efficiency measures of the penalization technique: error estimates of the control for L s norms (s in [1, +∞]), error estimates of the state and the adjoint state in Sobolev spaces W 1,s (s in [1, +∞)) and also error estimates for the value function. For the L 1 norm and the logarithmic penalty, the sharpest results are given, by establishing an error estimate for the penalized control of order ${O(\varepsilon|\log\epsilon|)}$ where ${\varepsilon >0 }$ is the (small) penalty parameter. 相似文献
10.
An Obstacle Control Problem with a Source Term 总被引:1,自引:0,他引:1
Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control,
and the goal is to keep the solution of the variational inequality close to the desired profile while the H
1
norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for
the existence of a minimizer. 相似文献
11.
Mohammed Benharrat Hocine Mokhtar-Kharroubi 《Rendiconti del Circolo Matematico di Palermo》2010,59(3):389-403
We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exterior penalty function to transform the constrained optimal control problem into a sequence of unconstrained optimal control problems, under conditions in control lie in L 1, the sequence of the solution to the unconstrained problem contains a subsequence converging of the solution of constrained problem, this convergence is strong when the problemis non convex, and is weak if the problemis convex in control. This generalizes the results of P.Nepomiastcthy [4] where he considered the control in the Hilbert space L 2(I,? m ). 相似文献
12.
M.C Delfour 《Journal of Mathematical Analysis and Applications》1977,60(1):8-35
In this paper we present a state theory for a class of linear functional differential equations of the retarded type considered by Delfour and Mitter (J. Differential Equations, 18 1975, 18–28) with initial functions in the product space Mp = X × Lp(?b, 0; X). Roughly speaking, the state at time t is a piece of trajectory defined over an interval [t ? b, t] for a fixed b > 0. From a study of the properties of the state in Mp an operational differential equation, the so-called state equation, is derived in order to describe its evolution. An adjoint state equation is also introduced for the adjoint state and the connection between solutions of the hereditary adjoint system and those of the adjoint state equation is established. All this provides the appropriate framework for the solution and the numerical approximation of the associated linear-quadratic optimal control and filtering problems. 相似文献
13.
T. Hou 《Numerical Analysis and Applications》2012,5(4):348-362
In this paper, we investigate the L ??(L 2)-error estimates and superconvergence of the semidiscrete mixed finite elementmethods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ?? 0). We derive error estimates for approximation of both state and control. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems. 相似文献
14.
Radouen Ghanem 《Positivity》2009,13(2):321-338
We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function
which is the control function is assumed in H2. We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions
existence and give necessary optimality conditions.
The author is grateful to Prof. M. Bergounioux for her instructive suggestions. 相似文献
15.
Carlos Conca Jeannine Saint Jean Paulin 《Journal of Mathematical Analysis and Applications》2003,285(1):17-36
The L2- and H1-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L2-approximate controllability or is locally distributed in the case of H1-approximate controllability. By using the classical Fenchel-Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem. 相似文献
16.
Zuliang Lu 《Applications of Mathematics》2016,61(2):135-163
We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L∞(J; L2Ω)-norm and L2(J; L2Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples. 相似文献
17.
Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control,
and the goal is to keep the solution of the variational inequality close to the desired profile while the H
1
norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for
the existence of a minimizer. 相似文献
18.
T. Hou 《Numerical Analysis and Applications》2013,6(2):163-175
In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h 2. Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results. 相似文献
19.
In this paper we study obstacle problems for the Navier (spectral) fractional Laplacian (?ΔΩ) s of order s ∈ (0,1) in a bounded domain Ω ? R n . 相似文献
20.
In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R2 as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R2 where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C2). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory. 相似文献