Optimal Control of a Multi-field Irrigation Problem: validation of a numerical solution by the optimality conditions


Optimal Control of a Multi-field Irrigation Problem:
validation of a numerical solution by the optimality

conditions
S.O. Lopesa, F.A.C.C. Fontesb

aCMAT and Departamento de Matemática e Aplicações, Universidade do Minho, Portugal
bSystec-ISR, Faculdade de Engenharia, Universidade do Porto, Portugal

sofialopes@math.uminho.pt, faf@fe.up.pt

Abstract—In this paper, we address the problem of minimizing
the total water consumption over the period of a year used to
supply different fields with different types of crops. We start
by recalling a previous study, where the authors developed
an optimal control model for this problem by minimizing the
water flowing into a reservoir and where the water from the
precipitation can be collected. The numerical solution obtained
using such model is analyzed. The main results in this paper
are the theoretical validation of the numerical solution via a
verifications that such solution satisfies the necessary conditions
of optimality in the form of a Maximum Principle. This way, we
are giving further evidence of the optimality of the numerical
solution found.

Keywords: Irrigation, Optimality conditions, Optimal
Control, Maximum Principle.

I. INTRODUCTION
Climate change is a reality. Recent findings from NASA

[17] report ”Globally-averaged temperatures in 2016 were
1.78 degrees Fahrenheit (0.99 degrees Celsius) warmer than
the mid-20th century mean. This makes 2016 the third year
in a row to set a new record for global average surface
temperatures. (...) NOAA scientists concur with the finding
that 2016 was the warmest year on record based on separate,
independent analyses of the data.”,

On the other hand, although our planet is blue, only 2.5%
of it is freshwater. From this 2.5% of freshwater, 70% is used
in agriculture.

According to [9], in 165 countries and 2 territories:
• the irrigation water requirement is 1500, 464 km3/yr
• the irrigation water total withdrawal is

2672, 640 km3/yr.
So, in this huge volume of a scarce resource, there is a waste
of 43.8% of water in the current irrigation systems.

Consequently, there is much that can be done to save water,
and it is of utmost important to our planet.

Naturally, the problem of minimizing water consumption in
irrigation systems has been the subject of several researcher
works, namely [1], [4] to mention a few using optimization
models. Using specifically optimal control techniques, that
take advantage of managing the storage of rainwater to save
water consunption, we mention the works [5], [20].

Fig. 1. The planet’s long-term warming trend is seen in this chart of every
year’s annual temperature cycle from 1880 to the present, compared to the
average temperature from 1880 to 2015. Record warm years are listed in the
column on the right. Credit: NASA/Earth Observatory.Joshua Stevens.

The authors have been studying the irrigation problem as an
optimal control problem. In a previous work [11], the authors
develop a model to minimize the water flowing into a reservoir
that supplies fields with different crops. In such work, only
the numerical solution was obtained, . Here, it is proved that
the numerical solution satisfies the necessary conditions of
optimality in the form of a Maximum Principle. Therefore, in
this paper the authors validate the numerical solution, adding
evidence of optimality to such solution.

The paper is organized as follows. Section 2 describes
formally the problem and the model used. Section 3 presents
the numerical results obtained for the problem. In Section 4
the numerical solution is validated. Concluding remarks are
drawn in Section 5.

II. PROBLEM SETTING

In this section, we describe the problem of minimizing the
total water consumed in irrigation over a long period of time
(one year is considered here) to supply various cultivation
fields with different types of crops. The problem is modeled as
an optimal control problem, based on the recent work reported
in [13], being able to capture adequately not only the dynamics
of the evolution along time of the variables used, but also

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the constraints that represent physical restrictions as well as
constraints that guarantee the needs of heathy crops.

We start by defining the variables used in the model. The
controls are: v total water flow coming from the tap and uj
water flow introduced in field j via its irrigation system. The
states are: xj water in the soil of field j and y total of amount
of water stored in reservoir.

The aim is to minimize the total of water flow coming from
the tap to a reservoir,

min

∫ T
0

v(t)dt,

where v(t) denotes the total water flow coming from the tap
at time t. The reservoir supplies the multiple cultivated fields
where each field can have different areas and different cultures.
The water from the precipitation also can be collected into the
reservoir. Therefore, the variation of water in reservoir is given
by

ẏ(t) = v(t) −
P∑

j=1

Ajuj(t) + Cg(t),

where y represents total of amount of water stored in the
reservoir, Aj represents the area of each field j, Cg(t)
represents collected water in a certain area C coming from
the precipitation g in the time t, and uj is the water flow
introduced in field j via its irrigation system.

The variation of water in the soil is given by the hydro-
logical balance equation, that is, the variation of water in the
soil is equal to what enters (water via irrigation systems and
precipitation) minus the loss (evapotranspiration of each crop
hj and loss by deep percolation βxj(t), a percentage of water
that is in the soil). So,

ẋj(t) = uj + g(t) −hj(t) −βxj(t), ∀j = 1, . . . ,P

where xj water in the soil of field j.
We assume that each field has only one crop. In order to

ensure that the crop is in good state of conservation, the water
in the each field has to be sufficient to satisfy the hydric needs
of each crop (xmin), that is:

xj(t) ≥ xminj.
The physical limitations of the amount of water that comes

from a tap, the amount of water that comes from the irrigation
systems, and the reservoir are given, respectively, by:

y(t) ∈ [0,ymax]
uj(t) ∈ [0,Mj]
v(t) ∈ [0,

∑
j AjMj]

where ymax is the maximum quantity of water in the reservoir
and Mj is the maximum water flow coming from the tap in
each field.

We assume that at the initial time the humidity in the soil
of each field and the water in the reservoir are given. Also,
the water in the reservoir at the initial time and to the final
time are imposed to be equal. So, we assume that

xj(0) = x0j
y(0) = y(T) = y0

A. The model
In summary, the optimal control formulation for our prob-

lem (OCP) with P fields and final time T is

min

∫ T
0

v(t)dt

subject to:
ẋj(t) = −βxj(t) + uj + g(t) −hj(t)

a.e. t ∈ [0,T], ∀j = 1, . . . ,P

ẏ(t) = v(t) −
∑P

j=1 Ajuj(t) + Cg(t), a.e. t ∈ [0,T],

xj(t) ≥ xminj, ∀t ∈ [0,T], ∀j = 1, . . . ,P

y(t) ∈ [0,ymax], ∀t ∈ [0,T]

uj(t) ∈ [0,Mj], a.e. , ∀j = 1, . . . ,P

v(t) ∈ [0,
∑

j AjMj], a.e.

xj(0) = x0j, ∀j = 1, . . . ,P

y(0) = y(T) = y0.

III. NUMERICAL RESULTS
We use a direct method discretization to obtain the numer-

ical results for the optimal control problem by transcribing it
as a mathematical programming problem. For that, we start
by considering a finite time grid i = 1, . . . ,N + 1

xi = x(ti)
ui = u(ti)

where ti = (i− 1)h and h = T/N.
Using the Euler-type discretization, the differential equation

ẋ = f(t,x,u)

is approximated by

xi = xi−1 + hf(ti−1,xi−1,ui−1).

To implement this optimization problem, the fmincon func-
tion of Matlab is used with the algorithm “active set”, by
default and the parameter “Tolfun” is considered 1E − 6.

The rainfall is estimated using an average of 10 years data
of rainfall for each month of the year, the dates are collected
from Instituto Português do Mar e da Atmosfera ([10]).

The Pennman - Monteith methodology [22] is used to cal-
culate evapotranspiration of culture along the year, according
the following formulation:

ET(ti) = KcET0(ti),

where Kc is the culture coefficient for the evapotranspiration
and ET0 is the tabulated reference value of evapotranspiration
from [19] for the Lisbon region.

To simulate the problem, we assume that:

P = 3 T = 12

Mj = 10 m3/month for each fieldj y0 = 0.01 ×At
ymax = 0.05 ×At β = 15%

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where At is the total area, (for more details see [12]).
Here, we also consider three crops: wheat in 1000 m2, sugar

cane in 750 m2 and olive in 1250 m2. The precipitation water
is collected in an area of 200 m2.

For these three crops, we consider:

xmin = [0.033 0.021 0.032]

Kc = [0.825 0.95 0.5]

x0 = 5xmin

The results obtained are reported in Fig. 2 and 3, (see also
[12]):

As expected, the crops need incoming water between May
and September. The months when the water consumption is
higher are June, July. The crop that needs less water is olive
with the largest area 1250 m2.

Note that we are imposing the constraint that the water
volume in the reservoir at the initial time is equal to the water
at the final time. It can be seen that until April the water from
the precipitation is saved in the reservoir and until this month
the tap is not opened. The tap is closed in September. June
and July are the months when the irrigation takes the highest
value, and also when the water in the reservoir decreases.

IV. VALIDATION OF THE SOLUTION

Since the problem has inequality constraints that are active
for some period of time, it is not easy to get a complete
analytical solution. However, we can verify that the numerical
solution for the mathematical programming problem satisfies
the necessary condition of optimality in the normal form of
Maximum Principle (MP). By normal form, we mean that the
scalar multiplier associated with the objective function — here
called λ — is nonzero.

The normal forms of the MP are guaranteed to supply
non-trivial information, in the sense that they guarantee that
the objective function is taken into account when selecting
candidates to optimal processes.

There has been a growing interest and literature on strength-
ened forms of NCO for Optimal Control Problems (OCP). The
normality results reported in literature require different degrees
of regularity on the problem data [6], [18], [16], [3], [2], [7],
[8], [14], [15].

In Rampazzo and Vinter [18], the MP can be written with
λ = 1, if there exists a continuous feedback u = η(t,ξ) such
that

dh(ξ(t))

dt
= ht(t,ξ) + hx(t,ξ) ·f(t,ξ,η(t,ξ)) < −γ′ (1)

for some positive γ′, whenever (t,ξ) is close to the graph
of the optimal trajectory, x̄(·), and ξ is near to the state
constraint boundary. There should exist a control pulling the
state variable away from the state constraint boundary.

In the problem, we can find three inequality constraints,
where the respective function are:

h1(x) = xmin −x
h2(y) = −y
h3(y) = y −ymax.

Observing the numerical solution in Fig. 3, we note that the
trajectory y never touches the upper limit. So, the inequality
constraint y(t) ≤ ymax is not active for any time in [0,T]. In
this case, it is not necessary verify the constraint qualification
to ensure normality.

From (1), we write

h1xj(ξ1j(t)) ·f(t,ξ1j,η1j(t,ξ1j))
= −(η1j(t,ξ1j) + 4j(t,ξ1j)) ≤−γ1, (2)

where 4j(t,ξ1j) = g(t)−βξ1j . For a ξ1j on a neighbourhood
of x̄j , we can always choose η1j sufficiently large so that the
equation (2) is satisfied, as long as Mj > βx̄j(t) − g(t), a
condition we can impose with loss of generality.

Again, from (1), we have

h2y(ξ2(t)) ·g(t,ξ2,η2(t,ξ2))
= −(η2(t,ξ2) −

∑P
j=1 Ajuj(t) + Cg(t)) ≤−γ2, (3)

As long as condition ymax >
∑P

j=1 Ajūj(t)−Cg(t) holds,
equation (3) is satisfied. This condition can be made to hold
but adequately dimensioning the reservoir capacity. Thus,
the inward pointing condition (1) is satisfied and normality
follows.

The problem data is sufficiently regular and the standard
hypotheses under which the Maximum Principle holds are
satisfied. Since the normality is ensured, we can apply the
Maximum Principle with λ = 1, (for example from [21]).

Define Hamiltonian function:

H(t, (x,y), (p,r), (u,v),λ) =

(p,r) · (−βx + u + g(t) − h(t),v + A · u + Cg(t)) −λv

Assuming that ((x̄, ȳ), (ū, v̄)) is a minimizer for (OCP),
then ∃(p,r) ∈ W1,1([0, 1] : RnP × R) and (µj,νL,νU ) ∈
C∗(0, 1):

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Fig. 2. Crops need.

Fig. 3. Reservoir

−(ṗ(t), ṙ(t)) = H(x,y)(t, (x̄, ȳ), (q(t),w(t)), (ū, v̄), 1)

H(t, (x̄, ȳ), (q(t),w(t)), (ū, v̄), 1) =
max(u,v)∈[0,M]×[0,R] H(t, (x̄, ȳ), (q(t),w(t)), (u,v), 1) a.e.;

supp{µj}⊂{t ∈ [0,T] : xj(t) = xminj}

supp{νL}⊂{t ∈ [0,T] : y(t) = 0}

supp{νU}⊂{t ∈ [0,T] : y(t) = ymax}

(q(0),−w(0),−q(T),−w(T)) ∈
+
NC (x̄(0), ȳ(0), x̄(T), ȳ(T))

(4)

where:

q(t) =




p(t) −
∫
[0,t)

µ(ds), t ∈ [0,T)

p(T) −
∫
[0,T ]

µ(ds), t = T,

w(t) =




r(t) −
∫
[0,t)

νL(ds) +

∫
[0,t)

νU (ds), t ∈ [0,T)

r(T) −
∫
[0,T ]

νL(ds) +

∫
[0,T ]

νU (ds), t = T.

and C = {x0}×{y0}×RnP ×{y0}

From the Adjoint Equation, we obtain


ṗ(t) = βq(t)

ṙ(t) = 0
(5)

From the Weierstrass Condition, we get:

(q(t) − A) · (u − ū) + (w(t) − 1)(v(t) − v̄(t)) ≤ 0 (6)

Since the endpoints are fixed, except for the final endpoint
of x that is free, the Transversality Condition is simplified to

q(T) = 0.

Therefore, in this problem, the Maximum Principle can be
written as follows. If ((x̄, ȳ), (ū, v̄)) is a minimizer for (OCP),

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then ∃(p,r) ∈ W1,1([0, 1] : RnP × R) and (µj,νL,νU ) ∈
C∗(0, 1):


ṗ(t) = βq(t)

ṙ(t) = 0

(q(t) − A) · (u − ū) + (w(t) − 1)(v(t) − v̄(t)) ≤ 0

q(T) = 0

Furthermore, in the case where (ū, v̄) ∈]0, M[×]0,R[, the
Weierstrass Condition can be written as:{

q(t) = A(t)
w(t) = 1.

(7)

From Fig. 4, we can conclude that (ū, v̄) ∈]0, M[×]0,R[
in June, July and August, and therefore we can apply the
Weierstrass Condition equation in the form of (7) in these
months.

We can observe the multiplier associated with the crops state
are equal to the area of the fields of the corresponding crops
in the months mentioned above:

wheat multiplier := 1000
sugar canne multiplier := 750
olive multiplier := 1250.

We also observe in Fig, 5, that the tranversality condition
is satisfied: q(T) = 0.

We also can see that in June, July and August the multiplier
associated with the reservoir state is equal to 1.

We conclude that for (ū, v̄) ∈]0, M[×]0,R[ the numerical
solution fulfills the Weierstrass and remaining conditions of
the Maximum Principle in the normal form.

In summary, we conclude that:
• the constraint qualification is verified, consequently we

can ensure normality;
• for (ū, v̄) ∈]0, M[×]0,R[ the numerical solution satisfies

the Weierstrass Condition of the Maximum Principle in
the normal form.

• the numerical solution satisfies the transversality condi-
tion.

• the numerical solution satisfies the adjoint equation.

V. CONCLUSION

We have addressed an irrigation planning problem of mini-
mizing the water used to supply different fields with different
types of crops, sharing a common reservoir, which not only
serves as storage, but also is from where the rain fall water is
collected. We have used an optimal control model, which is ca-
pable of adequately representing the dynamics of the humidity
of the soil taking into account irrigation, evapotranspirations
and infiltration, as well as taking into account the physical and
problem requirement constraints.

We have carried out a detailed analysis of the results ob-
tained numerically, namely by establishing for the problem the

necessary conditions of optimality in the form of a maximum
principle and verifying that the numerical solution fulfills all
the necessary conditions.

In the absence of additional candidates for optimality (ex-
tremals satisfying the necessary conditions) in the neighbor-
hood of the solution analysed, we may conclude that such
solution is in fact locally optimal.

Acknowledgments.

This work was partially supported by NORTE Regional
Operational Program through Project NORTE–45–2015–02
STRIDE – Smart Cyberphysical, Mathematical, Computa-
tional and Power Engineering Research for Disruptive In-
novation in Production, Mobility, Health, and Ocean Sys-
tems and Technologies, as well as projects POCI–01-
0145-FEDER-006933 -SYSTEC–Research Center for Sys-
tems and Technologies - Institute of Systems and Robotics,
Universidade do Porto, UID/MAT/00013/2013 - CMAT -
Centro de Matemática da Universidade do Minho, and
PTDC/EEI-AUT/2933/2014, POCI-01-0145-FEDER-016858
TOCCATTA, funded by FEDER funds through COM-
PETE2020 - Programa Operacional Competitividade e
Internacionalização (POCI) and by national funds through
FCT - Fundação para a Ciência e a Tecnologia.

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