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1.
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
2.
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
3.
Thomas Slawig 《Journal of Differential Equations》2005,219(1):116-143
We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper, we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of Gâteaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover, necessary first-order optimality conditions are stated, and the existence of a solution of a class of control problems is shown. 相似文献
4.
We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem. 相似文献
5.
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
6.
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
7.
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
8.
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
9.
This note focuses on a viscoelastodynamic problem being subject to unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, the problem can be reduced to the pseudo-differential linear complementarity problem through Fourier analysis. We prove that this problem possesses a solution, which is obtained as the limit of a sequence of solutions of penalized problems and we establish that the energy losses are purely viscous. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
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
11.
We study optimal control problems for the linearized Boussinesq system when the control is supported on a submanifold of the
boundary of the domain. This type of problem belongs to the class of optimal control problems with measures as controls, which
has been studied recently by several authors. We are mainly interested in the optimality conditions for such problems. It
is known that the differentiability properties needed to obtain the optimality conditions are more demanding, in terms of
regularity of the data, than what is needed to prove the existence of optimal controls. Here we are able to derive the optimality
conditions by taking advantage of the particular structure of the controls. 相似文献
12.
Francesca Bucci 《Journal of Evolution Equations》2007,7(3):387-414
We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists
of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational
inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal
control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system
inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness
of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of
solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic
equations with Neumann boundary data. 相似文献
13.
Truong Xuan Duc Ha 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(3):1305-1323
We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting. 相似文献
14.
Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations 总被引:2,自引:0,他引:2
In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality
conditions in the form of an exact Pontryagin's minimum principle for distributed and boundary controls (which can be unbounded)
and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear
parabolic equations.
Accepted 17 March 1997 相似文献
15.
P. Acquistapace 《Journal of Mathematical Analysis and Applications》2005,310(1):262-277
We consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations. 相似文献
16.
G. V. Alekseev 《Differential Equations》2016,52(3):361-372
We consider control problems for the 3D Maxwell equations describing electromagnetic wave scattering in an unbounded inhomogeneous medium that contains a permeable isotropic obstacle with cloaking boundary. Such problems arise when studying cloaking problems by the optimization method. The boundary coefficient occurring in the impedance boundary condition plays the role of a control. We study the solvability of the control problem and derive optimality systems that describe necessary conditions for the extremum. By analyzing the constructed optimality systems, we justify sufficient conditions imposed on the input data providing the uniqueness and stability of optimal solutions. 相似文献
17.
We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix. 相似文献
18.
In this paper we consider a class of stochastic evolution equations arising from initial boundary value problems with both boundary and distributed noise. We prove existence and regularity of mild solutions. Then we consider a controlled version of the model and prove the existence of optimal controls and develop the necessary conditions of optimality for partially observed problems using relaxed controls. 相似文献
19.
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions. 相似文献
20.
This paper deal with optimal control problems for a non-stationary Stokes system. We study a simultaneous distributed-boundary optimal control problem with distributed observation. We prove the existence and uniqueness of a simultaneous optimal control and we give the first order optimality condition for this problem. We also consider a distributed optimal control problem and a boundary optimal control problem and we obtain estimations between the simultaneous optimal control and the optimal controls of these last ones. Finally, some regularity results are presented. 相似文献