Normal forms of the Maximum Principle can be established if the problem satisfies suitable constraint qualification [ 13 — 15 ]. In this section, we verify that constraint qualification in [ 14 ] is satisfied for our problem.
Optimal Control of Partial Differential Equations: Theory, Methods and Applications
Let function represent the unmaximized Hamiltonian function: We note that the function is mensurable. Proposition 1 existence of solution and normality. If and for all , then i there exits an admissible solution; ii there exists an optimal solution; iii the Maximum Principle can be written with.
Existence of an Admissible Solution. It is easier to see that an admissible solution to OCP is where. We note that since , then. Existence of an Optimal Solution. Let us verify the conditions of theorem To apply theorem The assumptions of theorem In Rampazzo and Vinter [ 14 ], the MP can be written with , if there exists continuous feedback such that for some positive , whenever is close to the graph of the optimal trajectory, , and is near to the state constraint boundary.
There should exist a control flow of water provided by the irrigation systems pulling the state variable away from the state constraint boundary this guarantees that the crop survives. In our problem and, from 8 , we write where. For on a neighbourhood of , we can always choose sufficiently large so that 9 is satisfied, as long as , a condition we can impose with loss of generality.
Thus the inward pointing condition 9 is satisfied and normality follows. Proposition 2 necessary conditions. If the pair is a minimizer for the OCP, then there exists an absolutely continuous function and such that where is defined as follows: A known form of the normal MP for smooth problems with state constraints [ 1 , 3 , 15 ] is as follows. Let be a minimizer for OCP. Then there exists an absolutely continuous function and such that Applying these conditions to our problem, we have. Proposition 3 characterization of the optimal solution.
Let be the optimal control to OCP and let be the multiplier associated with the dynamic function on the MP.
Then the bounded variation function in Proposition 2 satisfies. If , we have that, for all , If , we have that, for all , In the remaining case i. We will use all the above information to validate the numerical solution already presented in [ 7 ] of our problem. In order to obtain this numerical solution, we consider next a discretized version of our problem. In this section, we obtain the numerical solution to our problem using sequence of finite dimensional nonlinear programming problems.
From now on, we consider the following corresponding discrete-time model:.
Optimal Control of Partial Differential Equations: Theory, Methods and Applications
OCPN is as follows: The dynamic equation implements the water balance in the soil: The rainfall, evapotranspiration, and losses models are described next, as presented in [ 10 ]. We defined average using the year data rainfall for each month of the year; the rain monthly average is as follows: To create the possibility of different weather scenarios, the rain monthly average is multiplied by a precipitation factor.
That means where the precipitation factor allows us to consider a typical year if this factor is 1, a drought year if it is less than 1, and a rainy year if it is above 1. This model is based on rain monthly average, so it is interesting if we are solving the yearly problem, for instance, if we want to design a reservoir [ 6 ] that can provide the necessary amount of water to our culture. We used the Pennman-Monteith methodology [ 16 ] to calculate evapotranspiration of our culture along the year.
In order to do so, we use the following formulation: The evapotranspiration of our culture in Lisbon is given by the following table: That means the dynamical equation is where. Note that we have a first-order linear ordinary differential equation with integrating factor equal to. From 19 and 24 , one may say that , where depends on the type of soil. We consider a field of potatoes in the region of Lisbon, Portugal. To obtain the numerical solution for the optimal control problem we have approximated the problem by a sequence of finite dimensional nonlinear programming problems see [ 18 ].
The numerical solution and the expected multipliers are plotted in Figure 1. It can be seen that the value of the optimal amount of water in the soil stays at the minimum allowed value from June till September. The irrigation should start in May; the maximum value is in June and stops in September. The water needs are 0. The code produces results that are according to what is expected for this region [ 3 ].
We can observe that and since is never equal to , is never greater than , as expected from Section 3. From here, we can say that although the analytical explicit solution was not obtained, the numerical solution fulfils the necessary optimality conditions.
Taking these information into account we now get an analytical characterization of the solution and multiplier. Since the inequality constraint is not active, then. Thus we most have and since , by the adjoint equation of the MP, we can conclude that. Applying the Weierstrass condition of MP, we get ,. Replacing by zero in the dynamics, we have As , then , for. Therefore and , for.
Therefore And we may conclude that for. Since for , then by the MP we also conclude that ,.
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Again, as and , by the adjoint equation of the MP, we can conclude that. On the other hand ; then by Weierstrass condition we get ,. Consequently, our dynamics is written as follows: Since , then , for. In Figure 2 , we plot the numerical and analytical solution obtained. We confirm that the numerical solution agrees with the results shown in Section 3. The analytical and estimated results of trajectory, control, and multipliers coincide. In the previous section, we have computed optimal yearly planning for the needs of water.
However, such planning is open-loop, meaning that the input used the prediction for the rainfall along the year as known in the beginning of the year. As we advance through the year, knowledge of the effective past rainfall and a better prediction of the rainfall for the coming months is available.
In this section we propose to use this newly available knowledge by resolving the optimal control problem at each month using the new rainfall information, in a receding horizon framework. In fact, due to the unpredictability of weather conditions, the numerical model presented in the previous section may not provide the real needs of water for the crop.
If we have an atypical year, values obtained by the rainfall model may be a lot different from reality, meaning that there is a high probability that the results obtained may not properly describe the reality if we use as input in our previous model the effective rainfall in each month, the estimated irrigation needs may be higher than if we use as input the rainfall model proposed in Section 3. In Figure 3 we show the results for the years and which were severe drought years in Portugal. For the year , if we look at real data, one can see that the irrigation should start in the mid-April.
Using the rainfall as input, the irrigation should start in May. The total amount of water used estimated using the rainfall model is and the real amount of water needed would be. For the year if we look at real data, one can see that the irrigation should start in the beginning of March. Using the rainfall model as input, the irrigation should start in May.
The real amount of water needed would be. This is due to the unpredictability of the weather.
Functional Analysis, Calculus of Variations and Optimal by Francis Clarke
The rainfall was much different from the expected, and, as a consequence, in such a scenario the proposed model would fail. To overcome this drawback, we propose replan strategy: To replan the systems we use model predictive control techniques. The predictive control technique generates a feedback strategy by solving a sequence of open-loop optimal control problems. In terms of the problem, this means that the irrigation strategy is frequently recomputed replan every time taking into account the measured system variable previous pluviosity.
In our case, we determine the optimal solution based on the OCPN and then at every time step we recalculate a new dynamic based on real data for rainfall. We test the replan model for the last ten years and we observe that state constraint is violated in the years , , , , and For data of year , the result obtained is described in Figure 5.
To avoid violation of the state constraint because of use of real data an improved model had to be considered. Modelling , 54 , Marshall , Evaluating HIV prevention strategies for populations in key affected groups: The example of Cabo Verde, Int.
Functional Analysis, Calculus of Variations and Optimal Control
Public Health , 60 , Torres , Dynamics and optimal control of Ebola transmission, Math. Torres , Vaccination models and optimal control strategies to dengue, Math. Torres , Seasonality effects on dengue: Basic reproduction number, sensitivity analysis and optimal control, Math. Torres , Optimal control of a tuberculosis model with state and control delays, Math. Watmough , Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Kaldor , Relation between HIV viral load and infectiousness: A model-based analysis, The Lancet , , Egger, Progression and mortality of untreated HIV-positive individuals living in resource-limited settings: Download as PowerPoint slide.
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