Numerical Analysis and Computation Commons^™

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, Brennan S. Mccann

Doctoral Dissertations and Master's Theses

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive …

Numerical Simulations For Optimal Control Of A Cancer Cell Model With Delay, Jessica S. Lugo

Murray State Theses and Dissertations

Mathematical models are often created to analyze the complicated behavior of many physical systems. One such system is that of the interaction between cancer cells, the immune system, and various treatments such as chemotherapy, radiation, and immunotherapy. Using models that depict these relationships gives researchers insight on the dynamics of this complicated system and possibly ideas for improved treatment schedules.

The model presented here gives the relationship of cancer cells in development phases with immune cells and cycle-specific chemotherapy treatment. This model includes a constant delay term in the mitotic phase. Optimal control theory is used to minimize the cost …

Mathematical Modeling And Optimal Control Of Alternative Pest Management For Alfalfa Agroecosystems, Cara Sulyok

Mathematics Honors Papers

This project develops mathematical models and computer simulations for cost-effective and environmentally-safe strategies to minimize plant damage from pests with optimal biodiversity levels. The desired goals are to identify tradeoffs between costs, impacts, and outcomes using the enemies hypothesis and polyculture in farming. A mathematical model including twelve size- and time-dependent parameters was created using a system of non-linear differential equations. It was shown to accurately fit results from open-field experiments and thus predict outcomes for scenarios not covered by these experiments.

The focus is on the application to alfalfa agroecosystems where field experiments and data were conducted and provided …

On The Lqg Theory With Bounded Control, D. V. Iourtchenko, J. L. Menaldi, A. S. Bratus

Mathematics Faculty Research Publications

We consider a stochastic optimal control problem in the whole space, where the corresponding HJB equation is degenerate, with a quadratic running cost and coeffcients with linear growth. In this paper we provide a full mathematical details on the key estimate relating the asymptotic behavior of the solution as the space variable goes to infinite.

Modelling And Optimal Control For Nonlinear Multistage Dynamical System Of Microbial Fed-Batch Culture, Chongyang Liu, Zhaohua Gong, Enmin Feng, Hongchao Yin

Chongyang Liu

In this paper, we propose a new controlled multistage system to formulate the fed-batch culture process of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD) by regarding the feeding rate of glycerol as a control function. Compared with the previous systems, this system doesn’t take the feeding process as an impulsive form, but a time-continuous process, which is much closer to the actual culture process. Some properties of the above dynamical system are then proved. To maximize the concentration of 1,3-PD at the terminal time, we develop an optimal control model subject to our proposed controlled multistage system and continuous state inequality constraints. …

Optimal Control For Nonlinear Dynamical System Of Microbial Fed-Batch Culture, Chongyang Liu

Chongyang Liu

In fed-batch culture of glycerol bio-dissimilation to 1, 3-propanediol (1, 3-PD), the aim of adding glycerol is to obtain as much 1, 3-PD as possible. So a proper feeding rate is required during the process. Taking the concentration of 1, 3-PD at the terminal time as the performance index and the feeding rate of glycerol as the control function, we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non-stationary control. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand we …

Optimal Switching Control For Microbial Fed-Batch Culture, Chongyang Liu, Zhaohua Gong, Enmin Feng, Hongchao Yin

Chongyang Liu

In fed-batch culture of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Hence, a proper feed strategy is required during the process. In this paper, we present an optimal switching control model based on our proposed controlled switching system. Some properties of the controlled switching system are obtained. Subsequently, we prove the existence of optimal control. In order to deduce the optimality conditions, we transcribe the optimal switching control model into an equivalent one with fixed switching instants and parameters. Finally, the optimality conditions of the equivalent problem are investigated …

Discrete Maximum Principle For Nonsmooth Optimal Control Problems With Delays, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

We consider optimal control problems for discrete-time systems with delays. The main goal is to derive necessary optimality conditions of the discrete maximum principle type in the case of nonsmooth minimizing functions. We obtain two independent forms of the discrete maximum principle with transversality conditions described in terms of subdifferentials and superdifferentials, respectively. The superdifferential form is new even for non-delayed systems and may be essentially stronger than a more conventional subdifferential form in some situations.

Optimal Control Of Stochastic Integrals And Hamilton-Jacobi-Bellman Equations, Ii, Pierre-Louis Lions, José-Luis Menaldi

Mathematics Faculty Research Publications

We consider the solution of a stochastic integral control problem, and we study its regularity. In particular, we characterize the optimal cost as the maximum solution of ∀v ∈ V, A(v)u ≤ ƒ(v) in D'(Ο), u = 0 on ∂Ο, u ∈ W^1,∞(Ο),

where A(v) is a uniformly elliptic second order operator and V is the set of the values of the control.

Optimal Control Of Stochastic Integrals And Hamilton-Jacobi-Bellman Equations, I, Pierre-Louis Lions, José-Luis Menaldi

Mathematics Faculty Research Publications

We consider the solution of a stochastic integral control problem and we study its regularity. In particular, we characterize the optimal cost as the maximum solution of ∀v ∈ V, A(v)u ≤ ƒ(v) in D'(Ο), u = 0 on ∂Ο, u ∈ W^1,∞(Ο),

where A(v) is a uniformly elliptic second order operator and V is the set of the values of the control.