A survey on numerical studies for fractional biological models and their optimal control

Mathematical models in biology are highly simplified representations of a complex underlying reality and there is always a high degree of uncertainty with regards to model function specification. This uncertainty becomes critical for models in which the use of different functions fitting the same dataset can yield substantially different predictions—a property known as structural sensitivity. Thus, even if the model is purely deterministic, then the uncertainty in the model functions carries through into uncertainty in model predictions, and new frameworks are required to tackle this fundamental problem. Here, we consider a framework that uses partially specified models in which some functions are not represented by a specific form. The main idea is to project infinite dimensional function space into a low-dimensional space taking into account biological constraints. The key question of how to carry out this projection has so far remained a serious mathematical challenge and hindered the use of partially specified models. Here, we propose and demonstrate a potentially powerful technique to perform such a projection by using optimal control theory to construct functions with the specified global properties. This approach opens up the prospect of a flexible and easy to use method to fulfil uncertainty analysis of biological models.

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Dynamics and optimal control of flexible solar updraft towers

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0539 ◽

2015 ◽

Vol 471 (2173) ◽

pp. 20140539 ◽

Cited By ~ 3

Author(s):

Michael Chi ◽

François Gay-Balmaz ◽

Vakhtang Putkaradze ◽

Peter Vorobieff

Keyword(s):

Optimal Control ◽

Three Dimensional ◽

Real Life ◽

Geometric Theory ◽

Two Dimensions ◽

Variable Pressure ◽

Numerical Studies ◽

Theory Of Optimal Control ◽

Analysis Of Dynamics ◽

Further Development

The use of solar chimneys for energy production was suggested more than 100 years ago. Unfortunately, this technology has not been realized on a commercial scale, in large part due to the high cost of erecting tall towers using traditional methods of construction. Recent works have suggested a radical decrease in tower cost by using an inflatable self-supported tower consisting of stacked toroidal bladders. While the statics deflections of such towers under constant wind have been investigated before, the key for further development of this technology lies in the analysis of dynamics, which is the main point of this paper. Using Lagrangian reduction by symmetry, we develop a fully three-dimensional theory of motion for such towers and study the tower's stability and dynamics. Next, we derive a geometric theory of optimal control for the tower dynamics using variable pressure inside the bladders and perform detailed analytical and numerical studies of the control in two dimensions. Finally, we report on the results of experiments demonstrating the remarkable stability of the tower in real-life conditions, showing good agreement with theoretical results.

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Solving the problems of aerospace aircraft optimal control

Engineering Journal Science and Innovation ◽

10.18698/2308-6033-2021-2-2058 ◽

2021 ◽

Author(s):

A.Yu. Melnikov

Keyword(s):

Optimal Control ◽

Analytical Calculation ◽

Initial Approximation ◽

Optimal Trajectories ◽

Physical Factors ◽

Aircraft Control ◽

Convergence Region ◽

The Maximum Principle ◽

Numerical Studies ◽

Automated Procedures

The paper overviews works on methods for solving problems of aerospace aircraft control and aims to solve the remaining problems. The purpose of the study was to obtain reference optimal control and its approximation in the guidance functions. When stating the problem, we took into account all essential physical factors; we selected the solution techniques and computational coordinate system, developed a mathematical model, methods and automated procedures for end-to-end optimization relying on the maximum principle, the formation and optimization of guidance functions. In addition to the well-known methods for solving boundary value problems, methods of analytical calculation of the initial approximation and the search for the convergence region were developed. The results of numerical studies contain nominal and perturbed optimal trajectories in which an aerospace aircraft is boosted by an accelerator, performs global spatial maneuvers, and enters orbit. Having analyzed extremals, compared and evaluated the effectiveness of the guidance functions, we drew certain conclusions taking into account the perturbations.

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