Pontryagin's Maximum Principle Applied to the Profile of a Beam

1967 ◽  
Vol 71 (679) ◽  
pp. 513-515 ◽  
Author(s):  
L. C. W. Dixon
Keyword(s):  
Maximum Principle ◽  
Lower Bound ◽  
Modulus Of Elasticity ◽  
Variable Density ◽  
Structural Member ◽  
Constant Density ◽  
Control Vector ◽  
The Maximum Principle ◽  
Centre Line ◽  
Independent Variable

Every structural member deforms under the forces acting upon it. In this paper the Maximum Principle, recently developed by Pontryagin is applied to the problem of determining that profile of a beam that deforms least under its own weight. The beam considered is a solid horizontal cantilever having constant density, modulus of elasticity and width. The height is taken as an independent variable subject to the restrictions that it lies between an upper and lower bound and is symmetrically distributed about the centre-line. The method is attractive in that it can readily be extended to take account of variable density, width and/or hollow members by extending the control vector u defined in the analysis.

scholarly journals Circuit Optimization Study According to the Maximum Principle

2021 ◽  
Vol 20 ◽  
pp. 362-371
Author(s):  
Alexander Zemliak
Keyword(s):  
Maximum Principle ◽  
Traditional Approach ◽  
Optimization Procedure ◽  
Circuit Optimization ◽  
Control Vector ◽  
Processor Time ◽  
The Maximum Principle ◽  
Cpu Time ◽  
Optimization Study ◽  
Time Minimization

The minimization of the processor time of designing can be formulated as a problem of time minimization for transitional process of dynamic system. A special control vector that changes the internal structure of the equations of optimization procedure serves as a principal tool for searching the best strategies with the minimal CPU time. In this case a well-known maximum principle of Pontryagin is the best theoretical approach for finding of the optimum structure of control vector. Practical approach for realization of the maximum principle is based on the analysis of behavior of a Hamiltonian for various strategies of optimization. The possibility of applying the maximum principle to the problem of optimization of electronic circuits is analyzed. It is shown that in spite of the fact that the problem of optimization is formulated as a nonlinear task, and the maximum principle in this case isn't a sufficient condition for obtaining a minimum of the functional, it is possible to obtain the decision in the form of local minima. The relative acceleration of the CPU time for the best strategy found by means of maximum principle compared with the traditional approach is equal two to three orders of magnitude.

Analysis of strategies of circuit optimisation on basis of maximum principle

2018 ◽  
Vol 37 (1) ◽  
pp. 484-503
Author(s):  
Alexander Zemliak
Keyword(s):  
Maximum Principle ◽  
Dimensional Case ◽  
Processing Unit ◽  
Local Minima ◽  
Control Vector ◽  
Content Type ◽  
The Maximum Principle ◽  
Central Processing ◽  
Cpu Time ◽  
First Time

Purpose This paper aims to propose a new approach on the problem of circuit optimisation by using the generalised optimisation methodology presented earlier. This approach is focused on the application of the maximum principle of Pontryagin for searching the best structure of a control vector providing the minimum central processing unit (CPU) time. Design/methodology/approach The process of circuit optimisation is defined mathematically as a controllable dynamical system with a control vector that changes the internal structure of the equations of the optimisation procedure. In this case, a well-known maximum principle of Pontryagin is the best theoretical approach for finding of the optimum structure of control vector. A practical approach for the realisation of the maximum principle is based on the analysis of the behaviour of a Hamiltonian for various strategies of optimisation and provides the possibility to find the optimum points of switching for the control vector. Findings It is shown that in spite of the fact that the maximum principle is not a sufficient condition for obtaining the global minimum for the non-linear problem, the decision can be obtained in the form of local minima. These local minima provide rather a low value of the CPU time. Numerical results were obtained for both a two-dimensional case and an N-dimensional case. Originality/value The possibility of the use of the maximum principle of Pontryagin to a problem of circuit optimisation is analysed systematically for the first time. The important result is the theoretical justification of formerly discovered effect of acceleration of the process of circuit optimisation.

scholarly journals Computer Time Minimizing for the Circuit Optimization

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Maximum Principle ◽  
Dynamic System ◽  
Traditional Approach ◽  
Optimization Procedure ◽  
Computer Time ◽  
Circuit Optimization ◽  
Control Vector ◽  
Processor Time ◽  
The Maximum Principle ◽  
Cpu Time

The solution of a problem of analogue circuit optimization is mathematically defined as a controllable dynamic system. In this context the minimization of the processor time of designing can be formulated as a problem of time minimization for transitional process of dynamic system. A special control vector that changes the internal structure of the equations of optimization procedure serves as a principal tool for searching the best strategies with the minimal CPU time. In this case a well-known maximum principle of Pontryagin is the best theoretical approach for finding of the optimum structure of control vector. Practical approach for realization of the maximum principle is based on the analysis of behaviour of a Hamiltonian for various strategies of optimization. It is shown that in spite of the fact that the problem of optimization is formulated as a nonlinear task, and the maximum principle in this case isn't a sufficient condition for obtaining a minimum of the functional, it is possible to obtain the decision in the form of local minima. The relative acceleration of the CPU time for the best strategy found by means of maximum principle compared with the traditional approach is equal two to three orders of magnitude.

A lower bound estimation for the Ricci curvature of a hypersurface in a hyperbolic space

2009 ◽  
Vol 46 (4) ◽  
pp. 539-546
Author(s):  
Mehmet Erdoğan
Keyword(s):  
Maximum Principle ◽  
Lower Bound ◽  
Hyperbolic Space ◽  
Ricci Curvature ◽  
Upper Bound ◽  
The Maximum Principle ◽  
Bound Estimation

In some previous papers the author gave an upper bound estimation for the Ricci curvature of a hypersurface in a hyperbolic space and in a sphere, see [4] and [5]. In the present paper, we give a lower bound estimation for the Ricci curvature of a compact connected embedded hypersurface in a hyperbolic space via the maximum principle given by H. Omori in [11].

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

Optimal Control of a General Environmental Space

1986 ◽  
Vol 108 (4) ◽  
pp. 330-339 ◽  
Author(s):  
M. A. Townsend ◽  
D. B. Cherchas ◽  
A. Abdelmessih
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Moisture Content ◽  
Control Strategies ◽  
Switching Control ◽  
The Maximum Principle ◽  
Environmental Space ◽  
And Performance

This study considers the optimal control of dry bulb temperature and moisture content in a single zone, to be accomplished in such a way as to be implementable in any zone of a multi-zone system. Optimality is determined in terms of appropriate cost and performance functions and subject to practical limits using the maximum principle. Several candidate optimal control strategies are investigated. It is shown that a bang-bang switching control which is theoretically periodic is a least cost practical control. In addition, specific attributes of this class of problem are explored.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES FOR A MODEL SYSTEM OF THE RADIATING GAS IN TWO DIMENSIONS

2008 ◽  
Vol 18 (04) ◽  
pp. 511-541 ◽  
Author(s):  
WENLIANG GAO ◽  
CHANGJIANG ZHU
Keyword(s):  
Cauchy Problem ◽  
Maximum Principle ◽  
Coupled System ◽  
Model System ◽  
Two Dimensions ◽  
Rarefaction Waves ◽  
Asymptotic Decay ◽  
The Maximum Principle ◽  
The Cauchy Problem ◽  
Asymptotic Decay Rate

In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L2-energy method, L1-estimate and the monotonicity of profile obtained by the maximum principle.

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