Consistency and feasibility of Haar wavelet collocation method for a

nonlinear optimal control problem with application

SAURABH R. MADANKAR1, AMIT SETIA1, MUNIYASAMY M.2, RAVI P. AGARWAL3

1Department of Mathematics, BITS Pilani K K Birla Goa Campus,

Zuarinagar, Goa-403726

INDIA

2Department of Mathematical and Computational Sciences, NITK,

Surathkal, Karnataka-575025

INDIA

3Department of Mathematics, Texas A&M University-Kingsville,

Kingsville, Texas 78363-8202,

USA

Abstract: Haar wavelet-based numerical algorithms have recently been developed for various mathematical prob-

lems, including optimal control problems. However, no numerical algorithm is complete without its theoretical

analysis. In this paper, we have shown the consistency and feasibility of the Haar wavelet-based collocation

method for solving nonlinear optimal control problems that have a single state and a single control variable with

constraints. The accuracy of the method has been shown through some application problems.

Key-Words: Haar wavelet, Optimal control, Feasibility, Consistency, Collocation method.

Received: April 15, 2023. Revised: December 19, 2023. Accepted: December 27, 2023. Published: December 31, 2023.

1 Introduction

Optimal control problems are applied in nearly all en-

gineering and scientific fields. It has many applica-

tions in robotics, aeronautics, the chemical, biochemi-

cal and medicine industry, etc. In chemical processes,

it helps to find an optimal policy that maximizes the

yield of a desired product. In the medical sector, re-

searchers use it to find an optimal amount of drug

dosage in people of different ages. It helps in the con-

trol of thermally unstable batch processes.

The optimal control problems have been solved

using various approaches, [1], in the literature, in-

cluding direct, indirect, and dynamic programming-

based methods. In 2020, [2], developed a collocation

method based on Legendre wavelet to deal with frac-

tional optimal control problems. To address optimal

control problems with non-smooth solutions, in 2021

the standard LGR collocation method, [3], is modi-

fied. The authors in [4], used an RBF collocation

method to address economic growth model optimal

control problems. The authors in [5], have used op-

timal control techniques to determine the effective-

ness of the oncolytic viral therapy for short-term treat-

ment. Using nonlinear delay differential equations

that simulate the spread of a computer virus. The au-

thors in [6], introduced Legendre-Gauss-Radau col-

location to approximate the optimal control prob-

lem. A technique for solving singular optimal con-

trol and bang-bang problems was created in 2022

by [7]. It involved the use of adaptive Legendre–

Gauss–Radau collocation. Most recently, the Volterra

integro-differential equation-governed optimal con-

trol problems were solved using a collocation method

based on Dickson polynomials, [8].

Haar wavelet-based numerical methods, [9], have

attracted significant attention of researchers in recent

years. The authors in [10], have highlighted some

of the advantages of the Haar wavelet method. The

authors in [11], developed a Haar wavelet colloca-

tion technique for boundary layer fluid flow prob-

lems in 2001. To solve elliptic partial differential

equations numerically, two novel and efficient ap-

proaches, [12], based on collocation with Haar and

Legendre wavelets were introduced in 2013. Two ef-

ficient methods based on collocation utilising Haar

and Legendre wavelets for the numerical solution of

linear as well as nonlinear differential equations are

proposed in 2014, [13]. A Haar wavelet-based ap-

proach was suggested by [14], in 2014 to solve a time

delayed optimal control problem. The convection-

diffusion equation can be solved more accurately,

simply, quickly, and computationally attractively us-

ing the Haar wavelet collocation approach, which was

developed in 2015 by [15]. In 2016, [16], developed

a Haar wavelet-based collocation algorithm and its

application to path planning and obstacle avoidance

problems.

In the next section, we define some preliminaries,

the Haar wavelets, and how these can be used in the

function approximation.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

652

Volume 18, 2023

2 Basic definitions and preliminaries

This section contains the introductory material and

the definitions used in this article.

Definition 2.1. Function space C1[0,1]

The space C1[0,1] is defined as the set of all contin-

uously differentiable real-valued functions on [0,1].

Definition 2.2. Function space L2[0,1]

The space L2[0,1] is defined by the set

L2[0,1] = f: [0,1] →R:Z1

0|f(t)|2dt < ∞,

with the norm defined as

∥f∥L2=Z1

0|f(t)|2dt

Definition 2.3. Haar wavelets

The Haar wavelet orthogonal family, [17], hi(t)de-

fined on [0,1) is defined for i= 0 as

h0(t) = 1, t ∈[0,1) ,

0,elsewhere,

and for i≥1as

hi(t) = 









1, t ∈hk

2j,k+1

2j,

−1, t ∈hk+1

2j,k+1

2j,

0,elsewhere,

where i= 2j+k, j = 0,1, . . . , k = 0,1, . . . , 2j−1.

Here, jand krepresent the dilation and translation

parameters, respectively.

The Haar wavelet family {hi(t)}∞

i=0 forms an or-

thonormal basis for L2[0,1] as proved by [18]. Hence,

any function z∈L2[0,1] can be written as

z(t) = ∞

i=0

cihi(t),

where ci=R1

0z(t)hi(t)dt.

The approximation zM(t)of z(t)by considering the

first M= 2j−1terms is given by

z(t)≈zM(t) =

M-1

i=0

cihi(t).

Similarly, the approximation ˙zM(t)of ˙z(t)can be

given as

˙z(t)≈˙zM(t) =

M-1

i=0

dihi(t),

from which we can get the approximation for z(t)as

zM(t) =

M-1

i=0

M-1

j=0

dipi+1,j+1hi(t) + z(0),

where P= [pi,j ],1≤i, j ≤M, is the Haar wavelet

integration matrix, [19].

3 Error bounds at the collocation

points

In this section, using existing results in the litera-

ture, we have derived the error bounds for the func-

tion approximated by Haar wavelets at the collocation

points.

Theorem 3.1. Let z(t)∈C1[0,1] and let zM(t)be

the Haar wavelet approximation, [20], of z(t).

Then

∥z−zM∥L2≤K

√3M,

where Kis the bound for ˙z(t).

Remark 3.2. Let z: [0,1] −→ Rbe a function

having bounded third-order derivative and ¨zMbe the

Haar wavelet approximation of ¨z, then by Theorem

3.1, we have

∥¨z−¨zM∥L2≤k′

√3M,(1)

where k′is the bound for ...

z(t). Thus, we have,

˙zM(t) = Rt

0¨zM(s)ds + ˙z(0). Hence, we get

|˙z(t)−˙zM(t)|=



Zt

(¨z(s)−¨zM(s))ds



≤Zt

0|¨z(s)−¨zM(s)|ds

≤Z1

0|¨z(s)−¨zM(s)|2ds

=∥¨z−¨zM∥L2.

Thus, we have, for all t∈[0,1]

|˙z(t)−˙zM(t)| ≤ ∥¨z−¨zM∥L2.(2)

Thus, at the collocation points tj=j+1/2

M, j =

0,1,2, . . . , M −1, we have, from (1) and (2)

|˙z(tj)−˙zM(tj)| ≤ k′

√3M.

Again, from (1) and (2) , it follows that

∥˙z−˙zM∥L2≤k′

√3M.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

653

Volume 18, 2023

Using a similar argument, we get

|z(tj)−zM(tj)| ≤ k′

√3M,

∥z−zM∥L2≤k′

√3M.

(3)

4 Optimal control problem and its

corresponding discretization

In this section, the optimal control problem and its

discretization are discussed.

The continuous optimal control problem is to deter-

mine the state x∈L2[0,1] and control u∈L2[0,1],

that minimizes the cost functional

I[x, u] = Z1

F(x, u)dt +E(x(0), x(1)),(4)

with dynamics

˙x=f(x, u),(5)

endpoint conditions

ep(x(0), x(1)) = 0, p = 1,2, . . . , Ne,(6)

and mixed constraint

gq(x, u)≤0, q = 1,2, . . . , Ng.(7)

It is assumed that F:R×R−→ R,E:R×R−→

R,f:R×R−→ R, with

ep:R×R−→ R, p = 1,2, . . . , Ne,

gq:R×R−→ R, q = 1,2, . . . , Ng,

are Lipschitz continuous with respect to each

argument. Additionally, it is assumed that an optimal

solution (x∗, u∗)exists.

Next, the discretized version of the continuous

optimal control problem (4)-(7) by using Haar

wavelet-based collocation method is described be-

low:

Determine xMand uMthat minimize

I[xM, uM] = 1

M−1

j=0

F(xM(tj), uM(tj))

+E(xM(0), xM(1)),

(8)

subject to

|˙xM(tj)−f(xM(tj), uM(tj))| ≤ δ1,

j= 0,1, . . . , M −1,

(9)

for each p= 1,2, . . . , Np

|ep(xM(0), xM(1))| ≤ δ2,(10)

for each q= 1,2, . . . , Ng

gq(xM(tj),(uM(tj)) ≤δ3, j = 0,1, . . . , M −1,

(11)

where δ1, δ2and δ3are the relaxation bounds, which

are positive constants dependent on M.

5 Feasibility and consistency of the

approximation

This section contains two results, the feasibility and

consistency of the Haar wavelet approximation (8)-

(11).

Theorem 5.1. Let (x(t), u(t)) be any given feasible

solution to the problem (4)-(7) such that x∈L2[0,1]

has third order bounded derivative and u∈L2[0,1]

has second order bounded derivative, then the prob-

lem (8)-(11) has a Haar wavelet feasible solution

(xM, uM)such that for j= 0,1, . . . , M −1

|x(tj)−xM(tj)| ≤ N1

√3M,

|u(tj)−uM(tj)| ≤ N2

√3M,

where N1, N2>0are the bounds for the third-

order derivative of xand the second-order derivative

of urespectively.

Proof. Since xhas a third-order bounded derivative,

we have, from Theorem 3.1

∥¨x−¨xM∥L2≤N1

√3M,(12)

where ¨xMis the Haar wavelet approximation of ¨xand

N1is the bound for ...

Then, from the Remark 3.2 it follows that, for each

j= 0,1, . . . , M −1, we obtain

|x(tj)−xM(tj)| ≤ N1

√3M,

|u(tj)−uM(tj)| ≤ N2

√3M,

|˙x(tj)−˙xM(tj)| ≤ N1

√3M.

For the dynamic constraint, we get

|˙xM(tj)−f(xM(tj), uM(tj))|

≤ |˙x(tj)−˙xM(tj)|

+|f(x(tj), u(tj)) −f(xM(tj), uM(tj))|

≤N1

√3M+l1|x(tj)−xM(tj)|+l2|u(tj)−uM(tj)|

≤N1

√3M+l1

√3M+l2

√3M,

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

654

Volume 18, 2023

where l1and l2are Lipschitz constants of fwith re-

spect to xand u, respectively, which are independent

of M.

It follows that, for j= 0,1, . . . , M −1, we have



˙xM(tj)−f(xM(tj), uM(tj))

≤δ1,

where

δ1=N1+l1N1+l2N2

√3M.

For each of the endpoint conditions with p=

1,2, . . . , Ne, we get

|ep(xM(0), xM(1))|

=|ep(xM(0), xM(1)) −ep(x(0), x(1))|

≤l1p|x(0) −xM(0)|+l2p|x(1) −xM(1)|

≤(l1p+l2p)N1

√3M,

where l1pand l2pare Lipschitz constants.

Thus,

|ep(xM(0), xM(1))| ≤ δ2,

where, δ2= (l1p+l2p)N1

√3M.

For each of the path constraints gq, q = 1, . . . , Ng

gq(xM(tj), uM(tj))

≤gq(xM(tj), uM(tj)) −gq(x(tj), u(tj))

≤ |gq(xM(tj), uM(tj)) −gq(x(tj), u(tj))|

≤l′′

1q|x(tj)−xM(tj)|+l′′

2q|u(tj)−uM(tj)|

≤l′′

√3M+l′′

√3M,

where l′′

1qand l′′

2qare Lipschitz constants of gqfor x

and urespectively. Thus we have

gq(xM(tj), uM(tj)) ≤δ3, j = 0,1, . . . , M −1,

where

δ3=l′′

1qN1+l′′

2qN2

√3M.

Thus, (xM, uM)is a feasible solution to the problem

(8)-(11).

Remark 5.2. The set of feasible solutions to the prob-

lems (8)-(11) is non-empty as a consequence of The-

orem 5.1.

Lemma 5.3. Let (xM, uM)be any feasible solution

to problem (8)-(11) such that xMand uMconverge

uniformly to xfand uf, respectively, with ˙xfand uf

continuous on [0,1]. Then, (xf, uf)is a feasible so-

lution to problem (4)-(7).

Proof. To prove that (xf, uf)is a feasible solution to

problem (4)-(7), first we show that it satisfies the dy-

namic constraint (5). By the contradiction argument,

suppose that (xf, uf)does not satisfy the differential

equation (5). Then there exists some t′∈[0,1] such

that

|˙xf(t′)−f(xf(t′), uf(t′))|>0.(13)

Since the collocation points tj, j = 0,1, . . . , M −1

are dense in [0,1], there exists a sequence of indices

{jM}∞

M=0 such that

lim

M→∞

tjM=t′.

Thus, we have

|˙xf(t′)−f(xf(t′), uf(t′))|

=lim

M→∞ |˙xM(tjM)−f(xM(tjM), uM(tjM))|

≤lim

M→∞

δ1= 0,

which implies that

˙xf(t′)−f(xf(t′), uf(t′)) = 0,

a contradiction to our assumption.

Hence, we have

˙xf(t) = f(xf(t), u(t)).

By a similar argument, it follows that (xf, uf)satis-

fies the endpoint constraints (6) and path constraints

(7). Hence, (xf, uf)is a feasible solution to problem

(4)-(7).

Next, we prove the consistency of the approxima-

tion.

Theorem 5.4. Suppose that (x∗

M, u∗

M)is a solution to

problem (8)-(11) and there exist (˜x, ˜u)with ˙

˜x(t)and

˜u(t)continuous on [0,1] such that x∗

M(t)→˜x(t)and

u∗

M(t)→˜u(t)uniformly. Then

lim

M→∞

I[x∗

M, u∗

M] = I[˜x, ˜u],(14)

and (˜x, ˜u)is an optimal solution to problem (4)-(7).

Proof. The cost functional of the continuous problem

is given by

I[˜x, ˜u] = Z1

F(˜x(t),˜u(t))dt +E(˜x(0),˜x(1)),

while the cost functional of the approximation prob-

lem is

I[x∗

M, u∗

M] = 1

M−1

j=0

F(x∗

M(tj), u∗

M(tj))

+E(x∗

M(0), x∗

M(1)).

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

655

Volume 18, 2023

To prove (14), we first show that

F(˜x(t),˜u(t))dt =

lim

M→∞ 



M−1

j=0

F(x∗

M(tj), u∗

M(tj))

.

Since F(˜x(t),˜u(t)) is continuous in t, we have,

[21]

F(˜x(t),˜u(t))dt =lim

M→∞ 



M−1

j=0

F(˜x(tj),˜u(tj))

.

Thus, we have

F(˜x(t),˜u(t))dt =

lim

M→∞ 



M−1

j=0

F(x∗

M(tj), u∗

M(tj))

+

lim

M→∞ 



M−1

j=0

(F(˜x(tj),˜u(tj)) −F(x∗

M(tj), u∗

M(tj)))

.

Now by the Lipschitz continuity of F(x, u), we have

|F(˜x(tj),˜u(tj)) −F(x∗

M(tj), u∗

M(tj)|

≤L(|˜x(tj)−x∗

M(tj)|+|˜u(tj)−u∗

M(tj)|),

for some L > 0and for all 0≤j≤M−1. Since

x∗

M(t)→˜x(t)and u∗

M(t)→˜u(t)uniformly, we have

lim

M→∞ |˜x(tj)−x∗

M(tj)|= 0,

and lim

M→∞ |˜u(tj)−u∗

M(tj)|= 0.

for all j= 0,1, . . . , M −1. Thus,

lim

M→∞ 



M−1

j=0

(F(˜x(tj),˜u(tj)) −F(x∗

M(tj), u∗

M(tj)))



≤lim

M→∞

M−1

j=0

(|˜x(tj)−x∗

M(tj)|+|˜u(tj)−u∗

M(tj)|)

= 0.

Hence, we get

F(˜x(t),˜u(t))dt =lim

M→∞

M−1

j=0

F(x∗

M(tj), u∗

M(tj)).

(15)

It is straightforward to prove that

lim

M→∞

E(x∗

M(0), x∗

M(1)) = E(˜x(0),˜x(1)).(16)

Thus, from (15) and (16), we have

lim

M→∞

I[x∗

M, u∗

M] = I[˜x, ˜u].

Next, to show that (˜x, ˜u)is an optimal solution to

problem (4)-(7), we first show that it is a feasible solu-

tion. By Lemma 5.3, it follows that (˜x, ˜u)is a feasible

solution to the problem (4)-(7).

Finally, we prove that (˜x, ˜u)is an optimal solution to

problem (4)-(7) by using the contradiction argument.

Suppose (˜x, ˜u)is not optimal and there exist optimal

(ˆx, ˆu)so that

I[ˆx, ˆu]< I[˜x, ˜u].

Now by Theorem 5.1, (ˆxM,ˆuM)is a feasible solution

to problem (8)-(11). But since (x∗

M, u∗

M)is assumed

to be optimal solution to problem (8)-(11), we have

I[x∗

M, u∗

M]< I[ˆxM,ˆuM].

Letting M→ ∞, we get

I[˜x, ˜u]< I[ˆx, ˆu],

a contradiction.

Hence (˜x, ˜u)is an optimal solution to the problem (4)-

(7).

6 Applications

In this section, the Haar wavelet collocation method is

applied to problems in fluid dynamics and economics.

The accuracy of the method has been shown by eval-

uating the maximum absolute error L∞defined as

L∞=max

j=0,1,...,M−1|y(tj)−yM(tj)|.

For the examples where the exact solution is un-

known, M= 128 has been considered as exact, and

the L∞error has been calculated with respect to it.

Example 6.1. (Temperature control of CSTR)

Consider the temperature control problem, [22], of

a continuous-stirred tank reactor (CSTR) by cooling-

rate manipulation. The optimal control problem for-

mulation of the temperature is as follows

min I[x, u] = Z0.5

[(x(t)−1.3)2+µu2(t)]dt,

subject to ˙x= 1 −x(t) + ae−γ/x(t)−u(t),

x(0) = 1.5, x(0.5) = 1.3,

where a= 1000, γ = 10 and µ= 0.25.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

656

Volume 18, 2023

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.3

1.32

1.34

1.36

1.38

1.4

1.42

1.44

1.46

1.48

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

Fig. 1: Comparison of approximate states and con-

trols for various values of Min case of Example 6.1.

Table 1. Comparison of values of cost functional and

error for various values of Min case of Example 6.1

M I[x∗

M(·), u∗

M(·)] L∞for state L∞for control

4 0.0892 0.0061 0.1659

8 0.0885 0.0040 0.0711

16 0.0882 0.0020 0.0314

32 0.0880 0.0009 0.0131

64 0.0879 0.0003 0.0043

The Haar wavelet approximations for the state and

control variables for M= 4,8,16,32,64,128 are

shown in Figure 1. The values of the approximate

cost functional and the L∞error in the state and con-

trol variables decrease with the increasing value of M,

as shown in Table 1.

Example 6.2. (Isothermal reaction in the presence

of catalyst)

Consider an isothermal liquid-phase reaction,

[23], in a Continuous stirred tank reactor (CSTR)

A−→ Bin the presence of a solid catalyst. It is

required to find the u(t), which represents the time-

dependent volumetric throughput per unit reactor vol-

ume, that minimizes the deviation in the concentra-

tion xof species Aand ufrom the reference condi-

tion (xs, us), in a given time tf. The corresponding

optimal control problem can be formulated as

I[x, u] = Ztf

[(x−xs)2+ (u−us)2]dt,

with dynamics

˙x=u(xf−x)−kx2,

x(0) = x0,

where, xfis the xin the feed and kis the reaction rate

coefficient.

The parameters taken to get the numerical solution are

x0= 5 g/cm3,xs= 8 g/cm3,xf= 10 g/cm3,

us= 5 min−1,k= 10−3cm3/(g·min)and tf= 1

min.

For M= 4,8,16,32,64,128, the approximate

state and control variables are plotted and shown in

Figure 2. It can be seen that the approximation im-

proves as the value of Mincreases. The value of the

cost functional, L∞error in the state and control vari-

ables also decreases with increasing value of Mand

is shown in Table 2.

Table 2. Comparison of values of cost functional and

error for various values of Min case of Example 6.2

M I[x∗

M(·), u∗

M(·)] L∞for state L∞for control

4 2.4464 0.3273 0.0954

8 2.4450 0.1420 0.0431

16 2.4443 0.0634 0.0195

32 2.4442 0.0266 0.0083

64 2.4441 0.0088 0.0027

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5.5

6.5

7.5

8.5

9.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4.6

4.7

4.8

4.9

5.1

5.2

5.3

5.4

5.5

Fig. 2: Comparison of approximate states and con-

trols for various values of Min case of Example 6.2.

Example 6.3. (Product quality control via pH

value in a chemical reaction)

The pH value of a chemical reaction is an impor-

tant factor that decides the quality of the product of the

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

657

Volume 18, 2023

reaction. In this example, [24], we consider a chemi-

cal reaction over a fixed interval [0, T ]. An ingredient

with strength u(t)is added to the chemical reaction to

control the pH value x(t)at time t. The rate of change

in pH is assumed to be proportional to the current pH

value and the strength of the ingredient u. The dy-

namics of the reaction is given by

˙x(t) = αx(t) + βu(t), x(0) = x0,

where α, β ∈R+are known and x0is the initial

pH value.

The cost functional to be minimized for this model is

given by

I[x(·), u(·)] = 1

2ZT

(ax2(t) + u2(t))dt,

where RT

0x2(t)dt is the decrease in the yield due

to changes in pH, and the cost rate of maintaining the

strength uis proportional to u2.

The exact solution to this problem is given by

x∗(t) = 1

aβ [c1(r+α)ert −c2(r−α)e−rt],

u∗(t) = c1ert +c2e−rt,

where

r=pα2+aβ2, c1=aβx0

(r+α)+(r−α)e2rT ,

c2=−aβx0e2rT

(r+α)+(r−α)e2rT .

Table 3. Comparison of values of cost functional and

error for different values of Min case Example 6.3

M I[x∗

M(·), u∗

M(·)] L∞for state L∞for control

4 15.9187 0.0127 1.3826

8 15.7378 0.0073 0.5832

16 15.6924 0.0059 0.2565

32 15.6811 0.0030 0.1069

64 15.6782 0.0011 0.0352

128 15.6775 0.0001 0.0001

The numerical solution is obtained by taking α=

2,β= 0.7,x0= 2,a= 3 and T= 1. The optimal

cost is 15.6773.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.2

1.4

1.6

1.8

2.2

2.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

Fig. 3: Comparison of approximate states and con-

trols for various values of Min case of Example 6.3.

Figure 3 shows that as the level of approximation

increases with M= 4,8,16,32,64, and 128, the

Haar wavelet approximations for the state and con-

trol variables become more accurate. The values of

the estimated cost functional and the L∞error in the

state and control variables decrease with increasing

values of M, as shown in Table 3.

Example 6.4. (Various investment problems) In

this example, we consider three types of investment

problems, [25], [26], namely, the unbounded invest-

ment problem, the bounded investment problem, and

the minimum control effort investment problem. Let

x(t)denote the available capital, u(t)the gross capi-

tal expenditures, and ˙x(t)the variation in the capital

stock. In the case of unbounded investment, we want

to maximize the profit performance measure, which

can be formulated as an optimal control problem to

maximize

I[x, u] = Z1

0x−1

2u2dt,

with dynamics ˙x=u−δx,

x(0) = 0, x(1) = free,

(17)

where δis the depreciation rate.

The optimal solution to this problem is given by

x(t) = 1 −1

2et−1+1

2e−1e−t,

u(t) = 1 −et−1,

(18)

and the optimal cost is 0.0840.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

658

Volume 18, 2023

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 4: Comparison of approximate states and con-

trols for various values of Min case of unbounded

investment problem.

Table 4. Comparison of values of cost functional and

error for various values of Min case of unbounded

investment problem

M I[x∗

M(·), u∗

M(·)] L∞for state L∞for control

4 0.0824 0.0067 0.0063

8 0.0836 0.0018 0.0017

16 0.0839 0.0004 0.0004

32 0.0840 0.0001 0.0001

64 0.0840 0.0001 0.0000

The bounded investment problem can be formu-

lated to maximize

I[x, u] = Z1

02x−1

2u2dt,

with dynamics ˙x=u−δx,

x(0) = 0, x(1) = 0,

(19)

where the capital path is required to be zero at T= 1.

The analytical solution is given by

x(t) = 2 −2et+ 2e1−t

e+ 1 ,

u(t) = 2 −4et

e+ 1,

(20)

and the optimal cost is 0.1515.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

Fig. 5: Comparison of approximate states and con-

trols for various values of Min case of bounded in-

vestment problem.

Table 5. Comparison of values of cost functional and

error for various values of Min case of bounded in-

vestment problem

M I[x∗

M(·), u∗

M(·)] L∞for state L∞for control

4 0.1433 0.0140 0.0224

8 0.1495 0.0037 0.0061

16 0.1510 0.0009 0.0016

32 0.1514 0.0002 0.0004

64 0.1515 0.0000 0.0001

Finally, the minimum control effort investment

problem, so called because of the performance mea-

sure of type RT

0u2dt, is to maximize

I[x, u] = Z1

0−u2dt,

with dynamics ˙x=u−δx,

x(0) = 0, x(1) = 1.

(21)

The exact solution is given by

x(t) = e1−t−e1+t

1−e2,

u(t) = −2e1+t

1−e2,

and the optimal cost is 2.3130.

By using an optimal investment strategy and mini-

mizing the total cost in the functional, which is di-

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

659

Volume 18, 2023

rectly proportionate to the investments made, the ob-

jective is to accumulate a unit of capital over the in-

terval [0,1].

For all three problems, the rate of depreciation, δ, is

assumed to be 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.2

1.4

1.6

1.8

2.2

Fig. 6: Comparison of approximate states and con-

trols for various values of Min case of minimum con-

trol effort investment problem.

Table 6. Comparison of values of cost functional and

error for various values of Min case of minimum con-

trol effort investment problem

M I[x∗

M(·), u∗

M(·)] L∞for state L∞for control

4 2.3092 0.0063 0.0114

8 2.3120 0.0017 0.0031

16 2.3127 0.0004 0.0008

32 2.3129 0.0001 0.0002

64 3.3130 0.0000 0.0000

Figure 4, Figure 5 and Figure 6 show the Haar

wavelet approximations for M= 4,8,16,32,64

along with exact state and exact control for the un-

bounded investment problem, the bounded invest-

ment problem, and the minimum control effort invest-

ment problem, respectively. It can be seen through

the graph that the approximations get better with in-

creasing levels of approximation. Also, Table 4, Ta-

ble 5, and Table 6 show that the Haar wavelet cost ap-

proaches the optimal cost as the value of Mincreases

in all three cases.

7 Conclusion

The consistency and feasibility of the Haar wavelet

collocation algorithm are proved for a nonlinear opti-

mal control problem with mixed state and control con-

straints. Through the consistency result, it has been

shown that the finite-dimensional nonlinear program-

ming problem (8)-(11), consistently approximates the

infinite-dimensional continuous problem (4)-(7). The

implementation of the algorithm has been shown to

solve various problems in fluid dynamics and eco-

nomics. In the future, the current study can be ex-

tended further for optimal control problems with mul-

tiple states and multiple control varibles. The ex-

tended study can be applied to wide-ranging fields

like aerospace, robotics, healthcare, and more while

also incorporating machine learning, deep learning,

and reinforcement learning.

Acknowledgment:

The first author acknowledges the financial support

(UGC-Ref. No. : 1309/(CSIR-UGC NET JUNE

2019)) by the University Grants Commission, New

Delhi, India.

References:

[1] Alessandra Buratto, Luca Grosset, Maddalena

Muttoni, et al. Two different solution

techniques for an optimal control problem with

a stochastic switching time. WSEAS

TRANSACTIONS ON MATHEMATICS,

22:730–735, 2023.

[2] Nitin Kumar and Mani Mehra. Legendre

wavelet collocation method for fractional

optimal control problems with fractional bolza

cost. Numerical methods for partial differential

equations, 37(2):1693–1724, 2021.

[3] Joseph D Eide, William W Hager, and Anil V

Rao. Modified legendre–gauss–radau

collocation method for optimal control

problems with nonsmooth solutions. Journal of

Optimization Theory and Applications, pages

1–34, 2021.

[4] Ahmad Golbabai, Nima Safaei, and

Mahboubeh Molavi-Arabshahi. Numerical

solution of optimal control problem for

economic growth model using rbf collocation

method. Computational Methods for

Differential Equations, 10(2):327–337, 2022.

[5] R Vithanage and SOPHIA RJ Jang. Optimal

immunotherapy of oncolytic viruses and

adopted cell transfer in cancer treatment.

WSEAS Trans. Biol. Biomed, 19:140–150,

2022.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

660

Volume 18, 2023

[6] Mehdi Shahini, Asyieh Ebrahimzadeh, and

Raheleh Khanduzi. A spectral collocation

method for computer virus spread case of

delayed optimal control problem. Bulletin of

the Iranian Mathematical Society,

48(2):507–535, 2022.

[7] Elisha R Pager and Anil V Rao. Method for

solving bang-bang and singular optimal control

problems using adaptive radau collocation.

Computational Optimization and Applications,

81(3):857–887, 2022.

[8] Maryam Alipour and Samaneh Soradi Zeid.

Optimal control of volterra integro-differential

equations based on interpolation polynomials

and collocation method. Computational

Methods for Differential Equations,

11(1):52–64, 2023.

[9] Bijil Prakash, Amit Setia, Shourya Bose, and

Ravi P Agarwal. Error analysis of a haar

wavelets-based numerical method with its

application to a nonlinear fractional dengue

model. International Journal of Computer

Mathematics, pages 1–19, 2022.

[10] G Hariharan, R Rajaraman, and K Kannan.

Haar wavelets approach of traveling wave

equation-a plausible solution of lightning

stroke model. International Journal of

Engineering & Technology, 2(2):149, 2013.

[11] Božidar Šarler, Imran Aziz, et al. Haar wavelet

collocation method for the numerical solution

of boundary layer fluid flow problems.

International Journal of Thermal Sciences,

50(5):686–697, 2011.

[12] Imran Aziz, Božidar Šarler, et al. Wavelets

collocation methods for the numerical solution

of elliptic bv problems. Applied Mathematical

Modelling, 37(3):676–694, 2013.

[13] H Kheiri and H Ghafouri. Haar and legendre

wavelets collocation methods for the numerical

solution of schrodinger and wave equations.

Acta Universitatis Apulensis, 37:01–14, 2014.

[14] Masoomeh Mansoori and Alireza Nazemi.

Solving infinite-horizon optimal control

problems of the time-delayed systems by haar

wavelet collocation method. Computational

and Applied Mathematics, 35:97–117, 2016.

[15] Inderdeep Singh and Sheo Kumar.

Approximate solution of convection-diffusion

equations using a haar wavelet method. Ital. J.

Pure Appl. Math, 35:143–154, 2015.

[16] Marziyeh Mortezaee and Alireza Nazemi. A

wavelet collocation scheme for solving some

optimal path planning problems. The ANZIAM

Journal, 57(4):461–481, 2016.

[17] Bijil Prakash, Amit Setia, and Shourya Bose.

Numerical solution for a system of fractional

differential equations with applications in fluid

dynamics and chemical engineering.

International Journal of Chemical Reactor

Engineering, 15(5):20170093, 2017.

[18] Alfred Haar. Zur theorie der orthogonalen

funktionensysteme. Georg-August-Universitat,

Gottingen., 1909.

[19] Chi Fan Chen and Chi-Huang Hsiao. Haar

wavelet method for solving lumped and

distributed-parameter systems. IEE

Proceedings-Control Theory and Applications,

144(1):87–94, 1997.

[20] E Babolian and A Shahsavaran. Numerical

solution of nonlinear fredholm integral

equations of the second kind using haar

wavelets. Journal of Computational and

Applied Mathematics, 225(1):87–95, 2009.

[21] Alfio Quarteroni, Fausto Saleri, Paola

Gervasio, et al. Scientific computing with

MATLAB and Octave, volume 3. Springer,

2006.

[22] Morteza Pakdaman and Sohrab Effati.

Approximating the solution of optimal control

problems by fuzzy systems. Neural Processing

Letters, 43:667–686, 2016.

[23] Simant Ranjan Upreti. Optimal control for

chemical engineers. Taylor & Francis, 2013.

[24] Sebastian Aniţa, Viorel Arnăutu, Vincenzo

Capasso, and Vincenzo Capasso. An

introduction to optimal control problems in life

sciences and economics: From mathematical

models to numerical simulation with

MATLAB®, volume 2. Springer, 2011.

[25] Giovanni Romeo. Elements of Numerical

Mathematical Economics with Excel: Static

and Dynamic Optimization. Academic Press,

2019.

[26] Eric Naevdal. Solving continuous-time

optimal-control problems with a spreadsheet.

The Journal of Economic Education,

34(2):99–122, 2003.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

661

Volume 18, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

-Saurabh R. Madankar, carried out conceptualiza-

tion, formal analysis, methodology, validation, vi-

sualization, writing-original draft formal analysis,

methodology, writing-original draft and program-

ming.

-Amit Setia carried out the conceptualization, super-

vision, and writing review and editing.

-Muniyasamy M. carried out the conceptualization,

formal analysis and validation.

-Ravi P. Agarwal carried out the supervision and writ-

ing review and editing.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Conflicts of Interest

The authors have no conflicts of interest to

declare that are relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

The first author acknowledges the financial support

(UGC-Ref. No. : 1309/(CSIR-UGC NET JUNE

2019)) by the University Grants Commission, New

Delhi, India.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.66

Saurabh R. Madankar, Amit Setia,

Muniyasamy M., Ravi P. Agarwal

E-ISSN: 2224-2856

662

Volume 18, 2023