Pointwise Constraints Research Articles

Optimal control problems of BV trajectories with pointwise state constraints

AbstractThis paper deals with some optimal control problems governed by ordinary differential equations with Radon measures as data and with pointwise state constraints. We study the properties of the value function and obtain its characterization by means of an auxiliary control problem of absolutely continuous trajectories. For this, we use some known techniques of reparametrization and graph completion.We are also interested in the characterization of the value function as the unique constrained viscosity solution of a Hamilton-Jacobi equation with measurable time dependant Hamiltonians.

Minimal Effort Problems and Their Treatment by Semismooth Newton Methods

The paper introduces minimum effort control problems. These provide an answer to the question of the smallest possible control bound which still allows us to drive the system to a target within a fixed time T. This is a counterpart to the time optimal control problem which minimizes the time required to drive the system to the target, given a control bound. The problem is formulated as an optimal control problem with pointwise constraint on the control. The necessary conditions of optimality are derived by Lagrange multiplier theory. The semismooth Newton method is applied to a properly regularized problem. Well-posedness and superlinear convergence of the semismooth Newton method are proved for linear control systems under a controllability condition. Numerical results are presented for demonstrating the applicability and feasibility of the proposed method.

Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients

We prove existence of minimizers for the multiple integral $\int$Ω$\l(u(x),\rho_1(x,u(x)) $∇$u(x)) \rho_2(x,u(x)) dx $W1,1u∂(Ω),(*) where Ω$\subset\R^d$ is open bounded, $u:$Ω$\toR$ is in the Sobolev space u∂($*$)+W1,10(Ω), with boundary data $u_$∂$(\cdot)\in$W1,1(Ω)$\cap C^{0}$(Ω); and $\l:R$Χ$R^d\to[0,\infty]$ is superlinear $L\oxB$-measurable with $\rho_1(\cdot,\cdot),\rho_2(\cdot,\cdot)\in C^{0}($ΩΧ$R)$ both $>0$. One main feature of our result is the unusually weak assumption on the lagrangian: l**$(\cdot,\cdot)$ only has to be $lsc$ at $(\cdot,0)$, i.e. at zero gradient. Here l**$(s,\cdot)$ denotes the convex-closed hull of $\l(s,\cdot)$. We also treat the nonconvex case $\l(\cdot,\cdot)\ne$l**$(\cdot,\cdot)$, whenever a well-behaved relaxed minimizer is a priori known. Another main feature is that $\l(s,\xi)=\infty$ is freely allowed, even at zero gradient, so that (*) may be seen as the variational reformulation of optimal control problems involving implicit first-order nonsmooth scalar partial differential inclusions under state and gradient pointwise constraints. The general case $\int$Ω$L(x,u(x),$∇$u(x))$ is also treated, though with less natural hypotheses, but still allowing $L(x,\cdot,\xi)$ non-$lsc$ for $\xi\ne0$.

A Mixed Finite Element Scheme for Optimal Control Problems with Pointwise State Constraints

In this paper, we propose a mixed variational scheme for optimal control problems with point-wise state constraints, the main idea is to reformulate the optimal control problems to a constrained minimization problem involving only the state, which is characterized by a fourth order variational inequality. Then mixed form based on this fourth order variational inequality is formulated and a direct numerical algorithm is proposed without the optimality conditions of underlying optimal control problems. The a priori and a posteriori error estimates are proved for the mixed finite element scheme. Numerical experiments confirm the efficiency of the new strategy.

Optimal extended optical flow and statistical constraints: A result of convergence

This work was motivated by the necessity of defining and computing a solution ρ of a transport equation (1) ∂ t ρ + v ⋅ ∇ x ρ = f ( t , x , ρ ) . subject to a non-convex pointwise constraint ρ ∈ C . For when the velocity v is a regular function and when f is a Lipschitz function, in [9] sufficient conditions are given for the solution to the transport equation (15) to satisfy such a constraint. In this paper an algorithm for computing a relaxed solution is investigated.

A Semismooth Newton Method for $\mathrm{L}^1$ Data Fitting with Automatic Choice of Regularization Parameters and Noise Calibration

This paper considers the numerical solution of inverse problems with an $\mathrm{L}^1$ data fitting term, which is challenging due to the lack of differentiability of the objective functional. Utilizing convex duality, the problem is reformulated as minimizing a smooth functional with pointwise constraints, which can be efficiently solved using a semismooth Newton method. In order to achieve superlinear convergence, the dual problem requires additional regularization. For both the primal and the dual problems, the choice of the regularization parameters is crucial. We propose adaptive strategies for choosing these parameters. The regularization parameter in the primal formulation is chosen according to a balancing principle derived from the model function approach, whereas the one in the dual formulation is determined by a path-following strategy based on the structure of the optimality conditions. Several numerical experiments confirm the efficiency and robustness of the proposed method and adaptive strategy.

PDE-Constrained Optimization Subject to Pointwise Constraints on the Control, the State, and Its Derivative

A general Moreau–Yosida-based framework for minimization problems subject to partial differential equations and pointwise constraints on the control, the state, and its derivative is considered. A range space constraint qualification is used to argue existence of Lagrange multipliers and to derive a KKT-type system for characterizing first-order optimality of the unregularized problem. The theoretical framework is then used to develop a semismooth Newton algorithm in function space and to prove its locally superlinear convergence when solving the regularized problems. Further, for maintaining the local superlinear convergence in function space it is demonstrated that in some cases it might be necessary to add a lifting step to the Newton framework in order to bridge an $L^2$-$L^r$-norm gap, with $r>2$. The paper ends by a report on numerical tests.

Parametric dual regularization for an optimal control problem with pointwise state constraints

The perturbation method is used in the dual regularization theory for a linear convex optimal control problem with a strongly convex objective functional and pointwise state constraints understood as ones in L2. Primary attention is given to the qualitative properties of the dual regularization method, depending on the differential properties of the value function (S-function) in the optimization problem. It is shown that the convergence of the method is closely related to the Lagrange principle and the Pontryagin maximum principle. The dual regularization scheme is shown to provide a new method for proving the maximum principle in the problem with pointwise state constraints understood in L2 or C. The regularized Lagrange principle in nondifferential form and the regularized Pontryagin maximum principle are discussed. Illustrative examples are presented.

MAINTENANCE OF AGE-STRUCTURED POPULATIONS: OPTIMAL CONTROL, STATE CONSTRAINTS, AND BANG–BANG REGIME

The paper analyzes the optimal maintenance of a "fully manageable" biological population with no natural reproduction. Examples of such populations include dairy farms, horse breeding, gardening, and forestry. Mathematically, the problem leads to the nonlinear optimal control of a first-order partial differential equation with pointwise state constraints. A necessary extremum condition is proven using the abstract optimization theory in Banach spaces. It is shown that the optimal dynamics has a bang–bang structure with three jumps and involves buying new young individuals and selling the old ones when their anticipated future expenses exceed the profit.

Optimal extended optical flow subject to a statistical constraint

This work is motivated by the necessity to improve heart image tracking. This technique is related to the ability of generating an apparent continuous motion, which is observable through the variation of intensity from a starting image to an ending one. Given two images ρ 0 and ρ 1 , we calculate an evolution process ρ ( t , ⋅ ) which transports ρ 0 to ρ 1 by using the optimal extended optical flow. Such a strategy is found to be well suited for heart image tracking, provided the motion is controlled by a statistical model. In this paper we use viability theory to give sufficient conditions to handle the optimal extended optical flow subject to a point-wise statistical constraint by using Parzen’s approximation. The strategy is implemented in a 1D case and the numerical results which are presented show the efficiency of the proposed strategy.

Topological optimization of structures subject to Von Mises stress constraints

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in a structural topology design problem. However, according to the literature concerning the topological derivative, only the classical approach based on flexibility minimization for a given amount of material, without control on the stress level supported by the structural device, has been considered. In this paper, therefore, we introduce a class of penalty functionals that mimic a pointwise constraint on the Von Mises stress field. The associated topological derivative is obtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure is presented, but full justifications can be deduced from existing works. Then, a topology optimization algorithm based on these concepts is proposed, that allows for treating local stress criteria. Finally, this feature is shown through some numerical examples.

PDE-Constrained Optimization Subject to Pointwise Constraints on the Control, the State, and Its Derivative

A general Moreau-Yosida-based framework for minimization problems subject to partial differential equations and pointwise constraints on the control, the state, and its derivative is considered. A ...

Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems

Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented.

A penalty method for topology optimization subject to a pointwise state constraint

This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.

Lavrentiev-prox-regularization for optimal controlof PDEs with state constraints

A Lavrentiev prox-regularization method for optimal control problems with point-wise state constraints is introduced where both the objective function and the constraints are regularized. The convergence of the controls generated by the iterative Lavrentiev prox-regularization algorithm is studied. For a sequence of regularization parameters that converges to zero, strong convergence of the generated control sequence to the optimal control is proved. Due to the proxcharacter of the proposed regularization, the feasibility of the iterates for a given parameter can be improved compared with the non-prox Lavrentiev-Regularization.

Regularized state-constrained boundary optimal control of the Navier–Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier–Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau–Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.

Optimal Design in Conductivity Under Locally Constrained Heat Flux

This paper addresses the relaxation of an optimal design problem in conductivity under a point-wise constraint on the heat flux. By using a variational approach, developed by the authors in a previous work, we obtained a complete relaxation, involving the explicit computation of constrained quasiconvex envelopes. This relaxed formulation turns out to be very simple in the remarkable compliance case, for which numerical simulations are performed. Those simulations show that our approach and results are interesting from a theoretical point of view as well as from a practical perspective. The emphasis of our work is placed on the local constraint on the heat flux, which seems to be a new ingredient for this kind of optimal design problems.

Variational problems with pointwise constraints and degeneration in variable domains

In this article we deal with a sequence of functionals defined on weighted Sobolev spaces. The spaces are associated with a sequence of domains Ω s contained in a bounded domain Ω of R n . The main structural components of the functionals are integral functionals whose integrands satisfy a growth and coercivity condition with a weight and additional terms ψs ∈ L 1 (Ωs). For the given functionals we consider variational problems with sets of constraints for functions v of the kind h(x,v(x)) 0a .e. inΩs ,w hereh : Ω ×R → R. We establish conditions on h and ψs and on the given domains, weighted spaces and functionals under which solutions of the variational problems under consideration converge in a certain sense to a solution of a limit variational problem with the set of constraints defined by the same function h.

State-Constrained Optimal Control of Semilinear Elliptic Equations with Nonlocal Radiation Interface Conditions

We consider a control- and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The nonlocal radiation interface condition and the pointwise state constraints represent the particular features of this problem. To deal with the state constraints, continuity of the state is shown, which allows us to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an $L^2$-neighborhood.

Adaptive FEM for PDE Constrained Optimization with Pointwise Constraints on the Gradient of the State

AbstractWe are concerned with an interior penalty method for an optimal control problem with an elliptic PDE constraint and additional pointwise constraints on the gradient of the state variable. We will derive an estimator that gives qualitative information about the error in the cost functional due to the interior penalty parameter and for the discretization of the PDE. These can be used to balance the error contributions of both types of errors. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)