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

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## Abstract

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## 1. Introduction

**Analytic linear constraint optimization.**In this application, the analytical problem of finding the extreme of a quadratic vectorial form subject to linear constraint is obtained using ToC instead of using the classical Lagrange multipliers technique.**Brachistochrone problem.**This application is dedicated to one of the most famous problem in calculus of variations: the brachistochrone problem [11]. This problem is solved numerically and at machine error accuracy via the two-point boundary-value problem of the Euler-Lagrange equation associated with the brachistochrone functional integral definition. In particular, the solution is obtained by deriving the differential equation in polar coordinates.**Over-constrained differential equations.**This application takes into consideration the problem of solving ODEs subject to more constraints than the degree of the ODE which arise in many areas of science and engineering. For example, in the problem of orbit determination of a satellite the number of measurements exceeds the order of the governing differential equation [12]. Furthermore, multi-purpose optimization [13,14,15] deals specifically with optimizing across multiple objective functions simultaneous where trade-offs (weighting) between two conflicting objectives must be taken into account. In this section, a weight least-squares solution is provided for over-constrained differential equations by assigning relative weights to the constraints and then solving the weighted constrained ODE by ToC. Additionally, another example showing the sequence of continuous solutions of an IVP morphing into an BVP is provided.**Inequality constraints.**This application extends the ToC framework to include inequality constraints [16]. This is obtained using a combination of sigmoid functions to keep the constrained expression within the inequality constraints. This allows for the derivation of functions constrained by user-defined bounds than can be asymmetric, continuous, or symmetric discontinuous functions. The main motivation for this constraint type comes from optimal control problems (bounded control inputs).**Triangular domains.**Validated by mathematical proof, Ref. [2] has extended the original univariate theory [1] to the multivariate theory [2]. This extension represents the multivariate formulation of ToC subject to arbitrary-order derivative constraints in rectangular domains. This provides an analytical procedure to obtain constrained expressions in any orthogonal/rectangular space that can be used to transform constrained problems into unconstrained problems. In Ref. [2] particular emphasis and details are given to the 2-dimensional case, because of the most important applications to surfaces (PDEs, Topography, Visualization, etc.). This application begins the extension of ToC to triangular domains by deriving the surfaces satisfying boundary constraints.

## 2. Analytic Linear Constraints Optimization

## 3. Brachistochrone Problem

## 4. Over-Constrained Differential Equations

#### 4.1. Second Order Differential Equation with Three Point Constraints. A Numerical Example

#### 4.2. Initial to Boundary-Value Problem Transformation. A Numerical Example

## 5. Inequality Constraints

## 6. Triangular Domains

#### 6.1. Affine Transformation from the Unit Triangle to the Generic Triangle

#### 6.2. Coons-Type Surface on the Unit Triangle

#### 6.3. ToC Surfaces on the Unit Triangle

#### 6.4. ToC Surfaces on the Generic Triangle

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BVP | boundary-value problem |

DE | differential equation |

IVP | initial value problem |

ODE | ordinary differential equation |

PDE | partial differential equation |

QP | quadratic programming |

ToC | Theory of Connections |

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**Figure 1.**Brachistochrone Coordinate Systems: (

**a**) classic cartesian coordinate system, (

**b**) modified cartesian coordinate system, and (

**c**) polar coordinate system.

**Figure 3.**The classical formulation of (

**a**) interpolation and (

**b**) least-squares provides a single function solution. The ToC method provides the numerical framework to describe (

**c**) all possible functions satisfying the constraints and (

**d**) all possible functions of a least-squares fitting.

**Figure 4.**Monte Carlo test for 10,000 trials. Part (

**a**) represents the solutions of the differential equation over the varying observed “constraints” with part (

**b**) quantifying the accuracy of the solution. Parts (

**c**)–(

**e**) represent the distribution of the solution values compared with the true value of the “constraints”.

**Figure 6.**Overshoot of $y(x)$ for the upper bound constraint. This phenomenon can be quantified and can be used to eliminate the overshooting.

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**MDPI and ACS Style**

Johnston, H.; Leake, C.; Efendiev, Y.; Mortari, D.
Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding. *Mathematics* **2019**, *7*, 537.
https://doi.org/10.3390/math7060537

**AMA Style**

Johnston H, Leake C, Efendiev Y, Mortari D.
Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding. *Mathematics*. 2019; 7(6):537.
https://doi.org/10.3390/math7060537

**Chicago/Turabian Style**

Johnston, Hunter, Carl Leake, Yalchin Efendiev, and Daniele Mortari.
2019. "Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding" *Mathematics* 7, no. 6: 537.
https://doi.org/10.3390/math7060537