scholarly journals Optimal control of mixing in Stokes fluid flows

2007 ◽  
Vol 580 ◽  
pp. 261-281 ◽  
Author(s):  
GEORGE MATHEW ◽  
IGOR MEZIĆ ◽  
SYMEON GRIVOPOULOS ◽  
UMESH VAIDYA ◽  
LINDA PETZOLD
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Fluid Flows ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Optimal Controls ◽  
Microfluidic Mixing ◽  
Space Norm ◽  
Stokes Fluid

Motivated by the problem of microfluidic mixing, optimal control of advective mixing in Stokes fluid flows is considered. The velocity field is assumed to be induced by a finite set of spatially distributed force fields that can be modulated arbitrarily with time, and a passive material is advected by the flow. To quantify the degree of mixedness of a density field, we use a Sobolev space norm of negative index. We frame a finite-time optimal control problem for which we aim to find the modulation that achieves the best mixing for a fixed value of the action (the time integral of the kinetic energy of the fluid body) per unit mass. We derive the first-order necessary conditions for optimality that can be expressed as a two-point boundary value problem (TPBVP) and discuss some elementary properties that the optimal controls must satisfy. A conjugate gradient descent method is used to solve the optimal control problem and we present numerical results for two problems involving arrays of vortices. A comparison of the mixing performance shows that optimal aperiodic inputs give better results than sinusoidal inputs with the same energy.

scholarly journals Mean-field optimal control for biological pattern formation

2021 ◽  
Author(s):  
Martin Burger ◽  
Lisa Maria Kreusser ◽  
Claudia Totzeck
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Mean Field ◽  
Wasserstein Distance ◽  
Descent Method ◽  
Gradient Descent Method ◽  
First Order ◽  
First Order Optimality Conditions

We propose a mean-field optimal control problem for the parameter identification of a  given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation  and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level.  Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.

scholarly journals On linear-quadratic optimal control of implicit difference equations

2018 ◽  
Vol 36 (3) ◽  
pp. 779-833
Author(s):  
Daniel Bankmann ◽  
Matthias Voigt
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Difference Equations ◽  
Matrix Pencil ◽  
Spectral Structure ◽  
Optimal Controls ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Quadratic Optimal Control

Abstract In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur’e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur’e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

Optimal feedback control of stochastic McShane differential systems

1974 ◽  
Vol 11 (2) ◽  
pp. 302-309 ◽  
Author(s):  
N. U. Ahmed ◽  
K. L. Teo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Necessary Conditions ◽  
Parabolic Type ◽  
Optimal Feedback Control ◽  
Distributed Parameter ◽  
Optimal Controls ◽  
Necessary Conditions For Optimality ◽  
Reduced Problem

In this paper, the optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.

scholarly journals Free terminal time optimal control problem for the treatment of HIV infection

2016 ◽  
Vol 6 (1) ◽  
pp. 33-51 ◽  
Author(s):  
Amine Hamdache ◽  
Smahane Saadi ◽  
Ilias Elmouki
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Hiv Infection ◽  
Control Problem ◽  
Time Optimal Control ◽  
Optimal Controls ◽  
Terminal Time ◽  
Time Optimal ◽  
Time Optimal Control Problem ◽  
Free Terminal Time

In this work, an optimal control approach is presented in order to propose an optimal therapy for the treatment HIV infection using a combination of two appropriate treatment strategies. The optimal treatment duration and the optimal medications amount are considered. The main objective of this study is to be able to maximize the benet based on number of healthy CD4+ T-cells and CTL immune cells and to minimize the infection level and the overall treatment cost while optimizing the duration of therapy. The free terminal time optimal control problem is formulated and the Pontryagin's maximum principle is employedto provide the explicit formulations of the optimal controls. The corresponding optimality system with the additional transversality condition for the terminal time is derived and solved numerically using an adapted iterative method with a Runge-Kutta fourth order scheme and a gradient method routine.

scholarly journals On an optimal control problem involving second order hyperbolic systems with boundary controls

1983 ◽  
Vol 27 (1) ◽  
pp. 139-148 ◽  
Author(s):  
K.G. Choo ◽  
K.L. Teo ◽  
Z.S. Wu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Second Order ◽  
Optimal Controls ◽  
Convergence Properties ◽  
Boundary Controls ◽  
Necessary And Sufficient

In this paper, we consider an optimal control problem involving second-order hyperbolic systems with boundary controls. Necessary and sufficient conditions are derived and a result on the existence of optimal controls is obtained. Also, a computational algorithm which generated minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated.

Optimal feedback control of stochastic McShane differential systems

1974 ◽  
Vol 11 (02) ◽  
pp. 302-309
Author(s):  
N. U. Ahmed ◽  
K. L. Teo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Necessary Conditions ◽  
Parabolic Type ◽  
Optimal Feedback Control ◽  
Distributed Parameter ◽  
Optimal Controls ◽  
Necessary Conditions For Optimality ◽  
Reduced Problem

In this paper, the optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.

scholarly journals Optimal control for a mathematical model for chemotherapy with pharmacometrics

2020 ◽  
Vol 15 ◽  
pp. 69
Author(s):  
Maciej Leszczyński ◽  
Urszula Ledzewicz ◽  
Heinz Schättler
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Cancer Chemotherapy ◽  
Drug Effects ◽  
Optimal Controls ◽  
Angiogenic Therapy ◽  
Single Drug ◽  
Qualitative Changes

An optimal control problem for an abstract mathematical model for cancer chemotherapy is considered. The dynamics is for a single drug and includes pharmacodynamic (PD) and pharmacokinetic (PK) models. The aim is to point out qualitative changes in the structures of optimal controls that occur as these pharmacometric models are varied. This concerns (i) changes in the PD-model for the effectiveness of the drug (e.g., between a linear log-kill term and a non-linear Michaelis-Menten type Emax-model) and (ii) the question how the incorporation of a mathematical model for the pharmacokinetics of the drug effects optimal controls. The general results will be illustrated and discussed in the framework of a mathematical model for anti-angiogenic therapy.

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