On the existence of piecewise continuous optimal controls

1977 ◽  
Vol 20 (1) ◽  
pp. 31-37 ◽  
Author(s):  
E. S. Noussair
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Maximum Rate ◽  
Optimal Controls ◽  
Control Variables ◽  
Piecewise Continuous ◽  
The Cost

AbstractExistence of piecewise optimal control is proved when the cost function includes one or both of (a) a cost of sudden switching (discontinuity) of control variables, and (b) a cost associated with the maximum rate of variation of the control over segments of the path for which the control is continuous.

Optimal control of a diffusive eco-epidemiological predator–prey model

2020 ◽  
Vol 13 (07) ◽  
pp. 2050065
Author(s):  
Xuebing Zhang ◽  
Guanglan Wang ◽  
Honglan Zhu
Keyword(s):  
Optimal Control ◽  
Semigroup Theory ◽  
Prey Population ◽  
Optimal Controls ◽  
Controlled System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Cost ◽  
Infected Prey

In this study, we investigate the optimal control problem for a diffusion eco-epidemiological predator–prey model. We applied two controllers to this model. One is the separation control, which separates the uninfected prey from the infected prey population, and the other is used as a treatment control to decrease the mortality caused by the disease. Then, we propose an optimal problem to minimize the infected prey population at the final time and the cost cause by the controls. To do this, by the operator semigroup theory we prove the existence of the solution to the controlled system. Furthermore, we prove the existence of the optimal controls and obtain the first-order necessary optimality condition for the optimal controls. Finally, some numerical simulations are carried out to support the theoretical results.

scholarly journals Optimal Control of Virus Spread under Different Conditions of Resources Limitations

Information ◽  
2019 ◽  
Vol 10 (6) ◽  
pp. 214
Author(s):  
Paolo Di Giamberardino ◽  
Daniela Iacoviello
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Virus Spread ◽  
Human Virus ◽  
Positive Diagnosis ◽  
Control Actions ◽  
The People ◽  
Different Types ◽  
The Cost ◽  
Instantaneous Control

The paper addresses the problem of human virus spread reduction when the resources for the control actions are somehow limited. This kind of problem can be successfully solved in the framework of the optimal control theory, where the best solution, which minimizes a cost function while satisfying input constraints, can be provided. The problem is formulated in this contest for the case of the HIV/AIDS virus, making use of a model that considers two classes of susceptible subjects, the wise people and the people with incautious behaviours, and three classes of infected, the ones still not aware of their status, the pre-AIDS patients and the AIDS ones; the control actions are represented by an information campaign, to reduce the category of subjects with unwise behaviour, a test campaign, to reduce the number of subjects not aware of having the virus, and the medication on patients with a positive diagnosis. The cost function considered aims at reducing patients with positive diagnosis using as less resources as possible. Four different types of resources bounds are considered, divided into two classes: limitations on the instantaneous control and fixed total budgets. The optimal solutions are numerically computed, and the results of simulations performed are illustrated and compared to put in evidence the different behaviours of the control actions.

LEARNING AND RECOGNITION OF HUMAN ACTIONS USING OPTIMAL CONTROL PRIMITIVES

2009 ◽  
Vol 06 (03) ◽  
pp. 459-479
Author(s):  
SUMITRA GANESH ◽  
RUZENA BAJCSY
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Estimation Method ◽  
Nonlinear Least Squares ◽  
Performance Criterion ◽  
Human Motion ◽  
Human Actions ◽  
Mode Estimation ◽  
Nonlinear Least Squares Problem ◽  
The Cost

We propose a unified approach for recognition and learning of human actions, based on an optimal control model of human motion. In this model, the goals and preferences of the agent engaged in a particular action are encapsulated as a cost function or performance criterion, that is optimized to yield the details of the movement. The cost function is a compact, intuitive and flexible representation of the action. A parameterized form of the cost function is considered, wherein the structure reflects the goals of the actions, and the parameters determine the relative weighting of different terms. We show how the cost function parameters can be estimated from data by solving a nonlinear least squares problem. The parameter estimation method is tested on motion capture data for two different reaching actions and six different subjects. We show that the problem of action recognition in the context of this representation is similar to that of mode estimation in a hybrid system and can be solved using a particle filter if a receding horizon formulation of the optimal controller is adopted. We use the proposed approach to recognize different reaching actions from the 3D hand trajectory of subjects.

scholarly journals Inverse KKT: Learning cost functions of manipulation tasks from demonstrations

2017 ◽  
Vol 36 (13-14) ◽  
pp. 1474-1488 ◽  
Author(s):  
Peter Englert ◽  
Ngo Anh Vien ◽  
Marc Toussaint
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Cost Functions ◽  
Quadratic Program ◽  
Relevant Task ◽  
Kernel Hilbert Spaces ◽  
The Cost ◽  
Linear Parameterization

Inverse optimal control (IOC) assumes that demonstrations are the solution to an optimal control problem with unknown underlying costs, and extracts parameters of these underlying costs. We propose the framework of inverse Karush–Kuhn–Tucker (KKT), which assumes that the demonstrations fulfill the KKT conditions of an unknown underlying constrained optimization problem, and extracts parameters of this underlying problem. Using this we can exploit the latter to extract the relevant task spaces and parameters of a cost function for skills that involve contacts. For a typical linear parameterization of cost functions this reduces to a quadratic program, ensuring guaranteed and very efficient convergence, but we can deal also with arbitrary non-linear parameterizations of cost functions. We also present a non-parametric variant of inverse KKT that represents the cost function as a functional in reproducing kernel Hilbert spaces. The aim of our approach is to push learning from demonstration to more complex manipulation scenarios that include the interaction with objects and therefore the realization of contacts/constraints within the motion. We demonstrate the approach on manipulation tasks such as sliding a box, closing a drawer and opening a door.

scholarly journals Direct Variational Assimilation of Radar Reflectivity and Radial Velocity Data: Issues with Nonlinear Reflectivity Operator and Solutions

2020 ◽  
Vol 148 (4) ◽  
pp. 1483-1502 ◽  
Author(s):  
Chengsi Liu ◽  
Ming Xue ◽  
Rong Kong
Keyword(s):  
Radial Velocity ◽  
Cost Function ◽  
Radar Data ◽  
Radar Reflectivity ◽  
Control Variables ◽  
Mixing Ratios ◽  
Highly Nonlinear ◽  
The Cost ◽  
Lower Limits ◽  
Radial Velocity Data

Abstract Radar reflectivity (Z) data are either directly assimilated using 3DVar, 4DVar, or ensemble Kalman filter, or indirectly assimilated using, for example, cloud analysis that preretrieves hydrometeors from Z. When directly assimilating radar data variationally, issues related to the highly nonlinear Z operator arise that can cause nonconvergence and bad analyses. To alleviate the issues, treatments are proposed in this study and their performances are examined via observing system simulation experiments. They include the following: 1) When using hydrometeor mixing ratios as control variables (CVq), small background Z can cause extremely large cost function gradient. Lower limits are imposed on the mixing ratios (qLim treatment) or the equivalent reflectivity (ZeLim treatment) in Z observation operator. ZeLim is found to work better than qLim in terms of analysis accuracy and convergence speed. 2) With CVq, the assimilation of radial velocity (Vr) is ineffective when assimilated together with Z data due to the much smaller cost function gradient associated with Vr. A procedure (VrPass) that assimilates Vr data in a separate pass is found very helpful. 3) Using logarithmic hydrometeor mixing ratios as control variables (CVlogq) can also avoid extremely large cost function gradient, and has much faster convergence. However, spurious analysis increments can be created when transforming the analysis increments back to mixing ratios. A background smoothing and a lower limit are applied to the background mixing ratios, and are shown to be effective. Using CVlogq with associated treatments produces better reflectivity analysis that is much closer to the observation without resorting to multiple analysis passes, and the cost function minimization also converges faster. CVlogq is therefore recommended for variational radar data assimilation.

scholarly journals Sufficiency for Purely Essentially Bounded Optimal Controls

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 238
Author(s):  
Gerardo Sánchez Licea
Keyword(s):  
Nonlinear Dynamics ◽  
Optimal Control ◽  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Optimal Controls ◽  
Embedding Theorems ◽  
State Control ◽  
Discontinuous Solutions ◽  
Piecewise Continuous

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded.

scholarly journals The reserve of joint torque determines movement coordination

2021 ◽  
Author(s):  
Germain Faity ◽  
Denis Mottet ◽  
Simon Pla ◽  
Jérôme Froger
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Degrees Of Freedom ◽  
Minimum Cost ◽  
Joint Torque ◽  
Movement Coordination ◽  
Trunk Motion ◽  
Additional Load ◽  
The Cost ◽  
Healthy Participants

AbstractHumans coordinate biomechanical degrees of freedom to perform tasks at minimum cost. When reaching a target from a seated position, the trunk-arm-forearm coordination moves the hand to the well-defined spatial goal, while typically minimising hand jerk and trunk motion. However, due to fatigue or stroke, people visibly move the trunk more, and it is unclear what cost can account for this. Here we show that people recruit their trunk when the torque at the shoulder is too close to the maximum. We asked 26 healthy participants to reach a target while seated and we found that the trunk contribution to hand displacement increases from 11% to 27% when an additional load is handled. By flexing and rotating the trunk, participants spontaneously increase the reserve of anti-gravitational torque at the shoulder from 25% to 40% of maximal voluntary torque. Our findings provide hints on how to include the reserve of torque in the cost function of optimal control models of human coordination in healthy fatigued persons or in stroke victims.

Use of Power Transform Mixing Ratios as Hydrometeor Control Variables for Direct Assimilation of Radar Reflectivity in GSI En3DVar and Tests with Five Convective Storms Cases

2020 ◽  
Author(s):  
Lianglyu Chen ◽  
Chengsi Liu ◽  
Ming Xue ◽  
Gang Zhao ◽  
Rong Kong ◽  
...  
Keyword(s):  
Cost Function ◽  
Radar Data ◽  
Power Parameter ◽  
Control Variables ◽  
Convective Storms ◽  
Mixing Ratios ◽  
Variational Framework ◽  
Study Power ◽  
The Cost ◽  
Outbreak Case

AbstractWhen directly assimilating radar data within a variational framework using hydrometeor mixing ratios (q) as control variables (CVq), the gradient of the cost function becomes extremely large when background mixing ratio is close to zero. This significantly slows down minimization convergence and makes the assimilation of radial velocity and other observations ineffective because of the dominance of reflectivity observation term in the cost function gradient. Using logarithmic hydrometeor mixing ratios as control variables (CVlogq) can alleviate the problem but the high nonlinearity of logarithmic transformation can introduce spurious analysis increments into mixing ratios.In this study, power transform of hydrometeors is proposed to form new control variables (CVpq) where the nonlinearity of transformation can be adjusted by tuning exponent or power parameter p. The performance of assimilating radar data using CVpq is compared with those using CVq and CVlogq for the analyses and forecasts of five convective storm cases from spring of 2017. Results show that CVpq with p = 0.4 (CVpq0.4) gives the best reflectivity forecasts in terms of root mean square error and equitable threat score. Furthermore, CVpq0.4 has faster convergence of cost function minimization than CVq and produces less spurious analysis increment than CVlogq. Compared to CVq and CVlogq, CVpq0.4 have better skills of 0-3h composite reflectivity forecasts, and the updraft helicity tracks for the 16 May 2017 Texas and Oklahoma tornado outbreak case are more consistent with observations when using CVpq0.4.

scholarly journals Effect of Adding Hydrometeor Mixing Ratios Control Variables on Assimilating Radar Observations for the Analysis and Forecast of a Typhoon

Atmosphere ◽  
2019 ◽  
Vol 10 (7) ◽  
pp. 415 ◽  
Author(s):  
Dongmei Xu ◽  
Feifei Shen ◽  
Jinzhong Min
Keyword(s):  
Cost Function ◽  
Positive Impact ◽  
Precipitation Forecast ◽  
Track Forecast ◽  
Error Statistics ◽  
Control Variables ◽  
Background Error ◽  
Mixing Ratios ◽  
The Cost ◽  
Assimilation Scheme

The variational data assimilation (DA) method seeks the optimal analyses by minimizing a cost function with respect to control variables (CVs). CVs are extended in this study to include hydrometeor mixing ratios related variables besides the widely used sets of CVs (momentum fields, surface pressure, temperature, and pseudo-relative humidity). The impacts of the extra CVs are investigated in terms of hydrometeor mixing ratios to the assimilation of radar radial velocity (Vr) and reflectivity (RF) for the analysis and prediction of Typhoon Chanthu (2010). It is found that the background error statistics of the extended CVs from the National Meteorological Center (NMC) method is reliable. The track forecast is improved significantly by including hydrometeor mixing ratios as CVs to assimilate radar Vr and RF. The DA experiments using the hydrometer CVs show much improved intensity analysis and forecast. It also improves the precipitation forecast skills to some extent. The positive impact is significant using a direct RF assimilation scheme, when Vr and RF data are applied together. It suggests that when we applying an indirect RF assimilation scheme, the fitting of more hydrometers in the cost function will tend to cause a slight degradation for other variables such as the wind and temperature.

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