Wavelet-based approximations of pointwise bound constraints in Lebesgue and Sobolev spaces

2020 ◽  
Author(s):  
S Dahlke ◽  
T M Surowiec
Keyword(s):  
Convergence Rates ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Obstacle Problems ◽  
State Variables ◽  
Bound Constraints ◽  
Pointwise Bound ◽  
Single Scale ◽  
Alternative Means ◽  
Higher Regularity

Abstract Many problems in optimal control, PDE-constrained optimization and constrained variational problems include pointwise bound constraints on the feasible controls and state variables. Most well-known approaches for treating such pointwise inequality constraints in numerical methods rely on finite element discretizations and interpolation arguments. We propose an alternative means of discretizing pointwise bound constraints using a wavelet-based discretization. The main results show that the discrete, approximating sets converge in the sense of Mosco to the original sets. In situations of higher regularity, convergence rates follow immediately from the underlying wavelet theory. The approach exploits the fact that one can easily transform between a given multiscale wavelet representation and single-scale representation with linear complexity. This allows, for example, a direct treatment of variational problems involving fractional operators, without the need for lifting techniques. We demonstrate this fact with several numerical examples of fractional obstacle problems.

An Input-Output Optimal Control Model under Uncertain Influence and its Solution

2012 ◽  
Vol 433-440 ◽  
pp. 2974-2979
Author(s):  
Shu Rong Li ◽  
Feng Wang ◽  
Xiao Yu He
Keyword(s):  
Optimal Control ◽  
Coefficient Matrix ◽  
Control Model ◽  
Chance Constrained Programming ◽  
State Variables ◽  
Input Output ◽  
Bound Constraints ◽  
Uncertain Model ◽  
Calculation Results ◽  
Constrained Programming

An input-output optimal control model is established under uncertain influence in environment. The objective function, terminal constraint of state variables and bound constraints of control variables are considered with fuzziness. The direct consumption coefficient matrix and investment coefficient matrix are regarded as stochastic variables. Membership function and chance constrained programming are applied to convert the uncertain model to a definite one. Penalty function and Particle Swarm Optimization are used to solve the model. The calculation results of an example demonstrate that the uncertain model has more practical value to decision makers compared to a definite one.

scholarly journals Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 537 ◽  
Author(s):  
Hunter Johnston ◽  
Carl Leake ◽  
Yalchin Efendiev ◽  
Daniele Mortari
Keyword(s):  
Differential Equations ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Linear Constraint ◽  
Mathematical Framework ◽  
Constrained Problems ◽  
Collocation Points ◽  
Constraint Embedding ◽  
New Applications ◽  
Unconstrained Problems

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

scholarly journals Online Implementation of Inequality Constraints Monitoring in Dynamical Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Nadia Zanzouri ◽  
Ramzi Ben Messaoud ◽  
Mekki Ksouri
Keyword(s):  
Dynamical Systems ◽  
Inequality Constraints ◽  
Practical Implementation ◽  
State Variables ◽  
Real Process ◽  
Online Implementation ◽  
Safety Specification ◽  
Parity Space ◽  
Physical Limitations ◽  
Residual Generation

This paper deals with fault detection in dynamical systems where the state variables evolutions are constrained by inequality constraints. The latter corresponds either to physical limitations or to safety specification. Two classical residual generation approaches are studied, namely, parity space and unknown input observer approaches, and are extended to monitor the inequality constraints. A practical implementation on a real process is performed and permits to validate the relevance of the proposed methods.

Applications of Primal-dual Interior Methods in Structural Optimization

2003 ◽  
Vol 3 (1) ◽  
pp. 159-176 ◽  
Author(s):  
Ronald H. W. Hoppe ◽  
Svetozara I. Petrova
Keyword(s):  
Structural Optimization ◽  
Optimization Problems ◽  
Null Space ◽  
Inequality Constraints ◽  
Composite Ceramic ◽  
Ceramic Materials ◽  
Design Parameters ◽  
State Variables ◽  
Primal Dual ◽  
Search Approach

AbstractWe are concerned with structural optimization problems for technological processes in material science that are described by partial differential equations. In particular, we consider the topology optimization of conductive media in high-power electronic devices described by Maxwell equations and the optimal design of composite ceramic materials by homogenization modeling. All these tasks lead to constrained nonconvex minimization problems with both equality and inequality constraints on the state variables and design parameters. After discretization by finite elements, we solve the discretized optimization problems by a primal-dual Newton interior-point method. Within a line-search approach, transforming iterations are applied with respect to the null space decomposition of the condensed primal-dual system to find the search direction. Some numerical experiments for the two applications are presented.

A Short Inroad to Optimized Compactification of Composite Neutronic Shields

2021 ◽  
Author(s):  
Nassar Haidar
Keyword(s):  
Nuclear Reactors ◽  
Inequality Constraints ◽  
Quality Criteria ◽  
Complex Problem ◽  
State Variables ◽  
State Equations ◽  
Optimal Sequence ◽  
Decision Variables ◽  
Equality And Inequality Constraints ◽  
Multilayered Composites

Abstract Compact neutronic shields for mobile nuclear reactors or accelerator-based neutron beams are known to be optimized multilayered composites. This paper is a simplified short inroad to the complex problem of optimizing the design of such shields when they attenuate a neutron beam to extremise certain quality criteria, in plane geometry, subject to equality and inequality constraints. In the equality constraints, the interfacial polychromatic neutron fluxes are solutions to course-mesh finite-difference holonomic state equations. The set of these interfacial fluxes act as state variables,while the set of layer thicknesses, or their poisoning (by added neutron absorbers) concentrations are decision variables. The entire procedure is then demonstrated to be reducible to standard Kuhn-Tucker semi-linear programming that may also lead robustly to an optimal sequence for these layers.

scholarly journals Necessary conditions for constrained distributed parameter systems with deviating argument

1996 ◽  
Vol 37 (4) ◽  
pp. 495-511
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Necessary Conditions ◽  
Inequality Constraints ◽  
Terminal State ◽  
Hyperbolic Partial Differential Equations ◽  
Separation Theorems ◽  
Distributed Parameter ◽  
State Variables ◽  
Deviating Argument ◽  
Distributed Parameters ◽  
Type Boundary

AbstractIn this paper we consider an optimal control problem governed by a system of nonlinear hyperbolic partial differential equations with deviating argument, Darboux-type boundary conditions and terminal state inequality constraints. The control variables are assumed to be measurable and the state variables are assumed to belong to a Sobolev space. We derive an integral representation of the increments of the functionals involved, and using separation theorems of functional analysis, obtain necessary conditions for optimality in the form of a Pontryagin maximum principle. The approach presented here applies equally well to other nonlinear constrained distributed parameters with deviating argument.

scholarly journals Point-to-Point Collision-Free Trajectory Planning for Mobile Manipulators

2016 ◽  
Vol 85 (3-4) ◽  
pp. 523-538 ◽  
Author(s):  
Grzegorz Pajak ◽  
Iwona Pajak
Keyword(s):  
Collision Avoidance ◽  
Trajectory Planning ◽  
Three Dimensional ◽  
Nonholonomic Constraints ◽  
Inequality Constraints ◽  
Mobile Manipulator ◽  
Local Perturbation ◽  
Mobile Manipulators ◽  
State Variables ◽  
Mechanical Constraints

AbstractThe collision-free trajectory planning method subject to control constraints for mobile manipulators is presented. The robot task is to move from the current configuration to a given final position in the workspace. The motions are planned in order to maximise an instantaneous manipulability measure to avoid manipulator singularities. Inequality constraints on state variables i.e. collision avoidance conditions and mechanical constraints are taken into consideration. The collision avoidance is accomplished by local perturbation of the mobile manipulator motion in the obstacles neighbourhood. The fulfilment of mechanical constraints is ensured by using a penalty function approach. The proposed method guarantees satisfying control limitations resulting from capabilities of robot actuators by applying the trajectory scaling approach. Nonholonomic constraints in a Pfaffian form are explicitly incorporated into the control algorithm. A computer example involving a mobile manipulator consisting of nonholonomic platform (2,0) class and 3DOF RPR type holonomic manipulator operating in a three-dimensional task space is also presented.

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