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1.
《Applied Mathematics and Optimization》2008,45(3):325-345
We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions,
we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the
optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with
respect to the positive cone are the techniques used to obtain our results. 相似文献
2.
The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.Dedicated to Professor Mitsuru Ikawa on the occasion of his 60th birthday 相似文献
3.
Lou 《Applied Mathematics and Optimization》2008,47(2):121-142
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. 相似文献
4.
Lou 《Applied Mathematics and Optimization》2003,47(2):121-142
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. 相似文献
5.
Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems 总被引:2,自引:0,他引:2
J. F. Bonnans 《Applied Mathematics and Optimization》1998,38(3):303-325
This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation,
namely, a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is
a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second-order
optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second-order sufficient
condition holds, we give a formula for the second-order expansion of the value of the problem as well as the directional derivative
of the optimal control, when the cost function is perturbed. Then we extend the theory of second-order optimality conditions
to the case of vector-valued controls when the feasible set is defined by local and smooth convex constraints. When the space
dimension n is greater than 3, the results are based on a two norms approach, involving spaces L
2
and L
s
, with s>n/2 .
Accepted 27 January 1997 相似文献
6.
This article considers the optimal control of nuclear fission reactors modeled by parabolic partial differential equations.
The neutrons are divided into fast and thermal groups with two equations describing their interaction and fission, while a
third equation describes the temperature in the reactor. The coefficient for the fission and absorption of the thermal neutron
is assumed to be controlled by a function through the use of control rods in the reactor. The object is to maintain a target
neutron flux shape, while a desired power level and adjustment costs are taken into consideration. A nonlinear optimality
system of six equations is deduced, characterizing the optimal control. An iterative procedure is shown to contract toward
the solution of the optimality system in small time intervals. The theory is extended to include the effect of other fission
products, leading to coupled ordinary and partial differential equations. Numerical experiments are also included, suggesting
directions for further research.
Accepted 13 January 1998 相似文献
7.
In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of
a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality
constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal
solutions of the problem.
Accepted 6 May 1997 相似文献
8.
Optimality systems for optimal control problems governed by elliptic variational inequalities are derived. Existence of appropriately defined Lagrange multipliers is proved. A primal—dual active set method is proposed to solve the optimality systems numerically. Examples with and without lack of strict complementarity are included. Accepted 5 March 1999 相似文献
9.
Y. Fujita 《Applied Mathematics and Optimization》2001,43(2):169-186
In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and
the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution
gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution
to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First,
we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control
explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.}
\amsclass{49L20, 35G20, 93E20.}
Accepted 11 September 2000. Online publication 16 January 2001. 相似文献
10.
Abstract. We formulate a robust optimal stopping-time problem for a state-space system and give the connection between various notions
of lower value function for the associated games (and storage function for the associated dissipative system) with solutions
of the appropriate variational inequality (VI) (the analogue of the Hamilton—Jacobi—Bellman—Isaacs equation for this setting).
We show that the stopping-time rule can be obtained by solving the VI in the viscosity sense and a positive definite supersolution
of the VI can be used for stability analysis. 相似文献
11.
Abstract. We formulate a robust optimal stopping-time problem for a state-space system and give the connection between various notions
of lower value function for the associated games (and storage function for the associated dissipative system) with solutions
of the appropriate variational inequality (VI) (the analogue of the Hamilton—Jacobi—Bellman—Isaacs equation for this setting).
We show that the stopping-time rule can be obtained by solving the VI in the viscosity sense and a positive definite supersolution
of the VI can be used for stability analysis. 相似文献
12.
D. C. Dobson 《Applied Mathematics and Optimization》1999,40(1):61-78
The problem of designing a periodic interface between two materials in such a way that time-harmonic waves diffracted from the interface have a specified far-field pattern is studied. A minimization problem for the interface is formulated, and it is shown that solutions of constrained bounded variation exist. The differentiability of the cost functional is then studied, with no restrictions on the smoothness of the interface. Some computational issues are discussed, and finally the results of some numerical experiments are presented. Accepted 3 February 1998 相似文献
13.
An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear
nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. The associated differential
Riccati equation is studied from the point of view of semigroup theory; it is shown to have a classical, explicitly represented
solution for very general final data; weighted H?lder regularity results for the optimal pair are deduced.
Accepted 10 December 1997 相似文献
14.
The problem of designing a periodic interface between two different materials, which gives rise to a specified far-field
diffraction pattern for a given incoming plane wave, is considered. The time harmonic waves are assumed to be TM (transverse
magnetic) polarized. The diffraction problem is modeled by a generalized Helmholtz equation with transparent boundary conditions.
In this paper the design problem is relaxed to include highly oscillatory profiles. Existence of an optimal design is established.
The principal method is based on the theory of homogenization for the model equation.
Accepted 31 May 2000. Online publication 26 February 2001. 相似文献
15.
An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. We give the proof of some result stated in [AT5], and in addition we prove uniqueness of the Riccati operator, provided its final datum is suitably regular. Accepted 14 October 1998 相似文献
16.
In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet
boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence
theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful
variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us
to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations,
we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions.
Accepted 7 June 1996 相似文献
17.
F. Da Lio 《Applied Mathematics and Optimization》2000,41(2):171-197
We study a class of infinite horizon control problems for nonlinear systems, which includes the Linear Quadratic (LQ) problem,
using the Dynamic Programming approach. Sufficient conditions for the regularity of the value function are given. The value
function is compared with sub- and supersolutions of the Bellman equation and a uniqueness theorem is proved for this equation
among locally Lipschitz functions bounded below. As an application it is shown that an optimal control for the LQ problem
is nearly optimal for a large class of small unbounded nonlinear and nonquadratic pertubations of the same problem.
Accepted 8 October 1998 相似文献
18.
This work is concerned with an approximation process for the identification of nonlinearities in the nonlinear periodic wave equation. It is based on the least-squares approach and on a splitting method. A numerical algorithm of gradient type and the numerical implementation are given. Accepted 14 January 2001. Online publication 20 June 2001. 相似文献
19.
Valery Agoshkov Paola Gervasio Alfio Quarteroni 《Mediterranean Journal of Mathematics》2006,3(2):147-176
New domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first and second
order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence
results are proved for the coupled problem, and the convergence of iterative processes is analyzed.
The work was supported by the Russian Foundation for Basic Research (04-01-00615) and it was partly carried out while the
first author was visiting the IACS at EPFL. 相似文献
20.
We study the positivity of the second shape derivative around an equilibrium for a 2-dimensional functional involving the perimeter of the shape and its the Dirichlet energy under volume constraint. We prove that, generally, convex equilibria lead to strictly positive second derivatives. We also exhibit some examples where strict positivity of the second order derivative holds at an equilibrium while existence of a minimum does not. 相似文献