共查询到20条相似文献,搜索用时 78 毫秒
1.
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
2.
This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence
of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation
as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions
of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints,
we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last
one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive
Pontryagin's principles for problems with pointwise state constraints.
Accepted 12 July 2001. Online publication 21 December 2001. 相似文献
3.
《Nonlinear Analysis: Theory, Methods & Applications》2011,74(10):3242-3260
The purpose of this paper is to propose and study a mathematical model and a boundary control problem associated to the miscible displacement of hydrogen through the porous anode of a PEM fuel cell. Throughout the paper, we study certain variational problems with a priori regularity properties of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. We consider a control problem that permits to determine the boundary distribution of the pressure which provides an optimal configuration for the temperature and for the concentration, as well. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our boundary control problem. The necessary conditions of optimality are derived by passing to the limit in the penalized optimality conditions. 相似文献
4.
J. Frédéric Bonnans Audrey Hermant 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009
This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control. 相似文献
5.
This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems
with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls
for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality
conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for
some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles
for the original problem.
Accepted 21 July 2001. Online publication 21 December 2001. 相似文献
6.
This paper is concerned with first order necessary optimality conditions for state constrained control problems in separable Banach spaces. Assuming inward pointing conditions on the constraint, we give a simple proof of Pontryagin maximum principle, relying on infinite dimensional neighboring feasible trajectories theorems proved in [20]. Further, we provide sufficient conditions guaranteeing normality of the maximum principle. We work in the abstract semigroup setting, but nevertheless we apply our results to several concrete models involving controlled PDEs. Pointwise state constraints (as positivity of the solutions) are allowed. 相似文献
7.
In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented. 相似文献
8.
F. Y?lmaz 《Journal of Computational and Applied Mathematics》2011,235(16):4839-4850
The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case. 相似文献
9.
A class of nonlinear elliptic optimal control problems with mixed control-state constraints arising, e.g., in Lavrentiev-type regularized state constrained optimal control is considered. Based on its first order necessary optimality conditions, a semismooth Newton method is proposed and its fast local convergence in function space as well as a mesh-independence principle for appropriate discretizations are proved. The paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing Lavrentiev-parameter. The latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the Lavrentiev-parameter. 相似文献
10.
We consider the identification problem of two operators having different properties for the systems governed by nonlinear evolution equations. For the identification problem, we show the existence of optimal solutions and present necessary optimality conditions. We illustrate the approach on two examples. 相似文献
11.
We prove an existence theorem of Lagrange multipliers for an abstract control problem in Banach spaces. This theorem may be applied to obtain optimality conditions for control problems governed by partial differential equations in the presence of pointwise state constraints. 相似文献
12.
Summary.
An optimal control problem
for impressed cathodic systems in electrochemistry is studied.
The control in this problem is the current
density on the anode. A matching objective functional is
considered. We first demonstrate the existence and uniqueness
of solutions for the governing partial differential equation
with a nonlinear boundary condition. We then prove
the existence of an optimal solution.
Next, we derive a necessary condition of optimality
and establish an optimality system of equations.
Finally, we define a finite element algorithm and
derive optimal error estimates.
Received
March 10, 1993 / Revised version received July 4, 1994 相似文献
13.
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
14.
In this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state-endpoints equality constraints. Using the first variation, a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the Dubois–Reymond Lemma to the optimal control setting and time scales. A second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions. The control problem is studied under a controllability assumption, and with or without the shift in the state variable. These two forms of the problem are shown to be equivalent. 相似文献
15.
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
16.
We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces. 相似文献
17.
《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
18.
We study viscosity solutions of Hamilton-Jacobi equations that arise in optimal control problems with unbounded controls and discontinuous Lagrangian. In our assumptions, the comparison principle will not hold, in general. We prove optimality principles that extend the scope of the results of [23] under very general assumptions, allowing unbounded controls. In particular, our results apply to calculus of variations problems under Tonelli type coercivity conditions. Optimality principles can be applied to obtain necessary and sufficient conditions for uniqueness in boundary value problems, and to characterize minimal and maximal solutions when uniqueness fails. We give examples of applications of our results in this direction. 相似文献
19.
《Optimization》2012,61(4-5):617-627
Without the need of a constraint qualification, we establish the necessary and sufficient optimality conditions for minimax fractional programming. Using these optimality conditions, we construct a mixed dual model which unifies the Mond–Weir dual, Wolfe dual and a parameter dual models. Several duality theorems are established. Consequently, this article partly solves the problem posed by Lai et al. [H.C. Lai, J.C. Liu and K. Tanaka (1999). Duality without a constraint qualification for minimax fractional programming. Journal of Optimization Theory and Applications, 101, 109–125.]. 相似文献
20.
Employing the optimality (necessary and sufficient) conditions of a nondifferentiable minimax programming problem in complex spaces, we formulate a one-parametric dual and a parameter free dual problems. On both dual problems, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem. 相似文献