scholarly journals Exponential turnpike property for fractional parabolic equations with non-zero exterior data

2020 ◽  
Author(s):  
Mahamadi WARMA ◽  
Sebastian Zamorano
Keyword(s):  
Optimal Control ◽  
Time Horizon ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Heat Equations ◽  
Optimal Controls ◽  
Optimal Solutions ◽  
Turnpike Property ◽  
Final Destination ◽  
Time Averages

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-\Delta)^s$ ( $0<s<1$ ). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.

Solving multiobjective optimal control problems using an improved scalarization method

2020 ◽  
Vol 37 (4) ◽  
pp. 1524-1547
Author(s):  
Gholam Hosein Askarirobati ◽  
Akbar Hashemi Borzabadi ◽  
Aghileh Heydari
Keyword(s):  
Optimal Control ◽  
Uniform Distribution ◽  
Optimal Control Problems ◽  
Pareto Frontier ◽  
Optimal Solution ◽  
Control Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Multiobjective Optimal Control

Abstract Detecting the Pareto optimal points on the Pareto frontier is one of the most important topics in multiobjective optimal control problems (MOCPs). This paper presents a scalarization technique to construct an approximate Pareto frontier of MOCPs, using an improved normal boundary intersection (NBI) scalarization strategy. For this purpose, MOCP is first discretized and then using a grid of weights, a sequence of single objective optimal control problems is solved to achieve a uniform distribution of Pareto optimal solutions on the Pareto frontier. The aim is to achieve a more even distribution of Pareto optimal solutions on the Pareto frontier and improve the efficiency of the algorithm. It is shown that in contrast to the NBI method, where Pareto optimality of solutions is not guaranteed, the obtained optimal solution of the scalarized problem is a Pareto optimal solution of the MOCP. Finally, the ability of the proposed method is evaluated and compared with other approaches using several practical MOCPs. The numerical results indicate that the proposed method is more efficient and provides more uniform distribution of solutions on the Pareto frontier than the other methods, such a weighted sum, normalized normal constraint and NBI.

Existence of optimal controls for viscous flow problems

1992 ◽  
Vol 439 (1905) ◽  
pp. 81-102 ◽  
Keyword(s):  
Optimal Control ◽  
Flow Control ◽  
Viscous Flow ◽  
Existence Theorem ◽  
General Class ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Optimal Controls ◽  
Flow Problems ◽  
Existence Of Optimal Controls

A class of optimal control problems in viscous flow is studied. Main result is the existence theorem for optimal control. Three typical flow control problems are formulated within this general class.

scholarly journals Optimal spatial patterns in feeding, fishing, and pollution

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hannes Uecker
Keyword(s):  
Spatial Patterns ◽  
Time Horizon ◽  
Optimal Control Problems ◽  
Steady States ◽  
Control Problems ◽  
Optimal Controls ◽  
Infinite Time ◽  
Distributed Optimal Control ◽  
Distributed Optimal Control Problems ◽  
Spatially Distributed

<p style='text-indent:20px;'>Infinite time horizon spatially distributed optimal control problems may show so–called optimal diffusion induced instabilities, which may lead to patterned optimal steady states, although the problem itself is completely homogeneous. Here we show that this can be considered as a generic phenomenon, in problems with scalar distributed states, by computing optimal spatial patterns and their canonical paths in three examples: optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package <inline-formula><tex-math id="M1">\begin{document}$\mathtt{pde2path} $\end{document}</tex-math></inline-formula> to first compute bifurcation diagrams of canonical steady states, and then time–dependent optimal controls to control the systems from some initial states to a target steady state as <inline-formula><tex-math id="M2">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>. We consider two setups: The case of discrete patches in space, which allows to gain intuition and to compute domains of attraction of canonical steady states, and the spatially continuous (PDE) case.</p>

Controlled Lotka-Volterra systems in the modeling of biomedical processes

2021 ◽  
Author(s):  
Evgenii Khailov ◽  
Nikolai Grigorenko ◽  
Ellina Grigorieva ◽  
Anna Klimenkova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Numerical Study ◽  
Treatment Strategies ◽  
Main Tool ◽  
Necessary Condition ◽  
Control Problems ◽  
Optimal Controls ◽  
Controlled Systems ◽  
Volterra Systems

This book is devoted to a consistent presentation of the recent results obtained by the authors related to controlled systems created based on the Lotka-Volterra competition model, as well as to theoretical and numerical study of the corresponding optimal control problems. These controlled systems describe various modern methods of treating blood cancers, and the optimal control problems stated for such systems, reflect the search for the optimal treatment strategies. The main tool of the theoretical analysis used in this book is the Pontryagin maximum principle - a necessary condition for optimality in optimal control problems. Possible types of the optimal blood cancer treatment - the optimal controls - are obtained as a result of analytical investigations and are confirmed by corresponding numerical calculations. This book can be used as a supplement text in courses of mathematical modeling for upper undergraduate and graduate students. It is our believe that this text will be of interest to all professors teaching such or similar courses as well as for everyone interested in modern optimal control theory and its biomedical applications.

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