# An Optimal Control Problem Governed by Nonlinear First Order Dynamic Equation on Time Scales

Authors: Jian-Ping Sun; Qiu-Yan Ren; Ya-Hong Zhao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we are concerned with a class of optimal control problem governed by nonlinear first order dynamic equation on time scales. By imposing some suitable conditions on the related functions, for any given control policy, we first obtain the existence of a unique solution for the nonlinear controlled system. Then, we study the existence of an optimal solution for the optimal control problem.


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## Body

## 1. Introduction

The theory of time scales was introduced by Hilger in [1] in order to unify discrete and continuous analysis. Some foundational definitions and results from the calculus on time scales will be defined in Section 2. For more details, one can see [2–4].In recent years, the calculus of variations and optimal control problems on time scales have attracted the attention of some researchers. For example, [5–8] discussed the calculus of variations on time scales and [9–12] studied some maximum principles on time scales, while [13–16] investigated the existence of optimal solutions or the necessary conditions of optimality for some optimal control problems on time scales.In 2017, Guo [17] studied the projective synchronization problem of a class of chaotic systems in arbitrary dimensions. Firstly, a necessary and sufficient condition for the existence of the projective synchronization problem was presented. Secondly, an algorithm was proposed to obtain all the solutions of the projective synchronization problem. Thirdly, a simple and physically implementable controller was designed to ensure the realization of the projective synchronization. Finally, some numerical examples were provided to verify the effectiveness and the validity of the proposed results.In 2020, Xu and Zhang [18] investigated general mean-field linear-quadratic (LQ) games of stochastic large-population system, where the individual diffusion coefficient could depend on both the state and the control of the agent, and the control weight in the cost functional could be indefinite. The asymptotic suboptimality property of the decentralized strategies for the LQ games was derived through the consistency condition. A pricing problem was also studied, for which the decentralized suboptimal price was obtained.Throughout this paper, we always assume thatT is a time scale, T>0 is fixed, 0,T∈T and σ2T=σT. For each interval I of ℝ, we denote by IT=I∩T.Suppose that there is a flock of sheep in a pasture. We consider the changes in the number of sheep during a time interval0,σTT. It is well known that the supply of herbage, which influences growth rate and reproductive ability of sheep, is one of the main ways to control the number of sheep. Now, we define some related functions as follows:xt is the number of sheep at time trt is the number of births per unit of time at time tpt is the number of sales per sheep per unit of time at time tut is the amount of herbage supplied at time tqt is the number of sheep converted by per unit of herbage supplied per unit of time at time tLetUad be the admissible control set. Then, for any given control policy u∈Uad, it is easy to know that the changes in the number of sheep can be described by the following linear dynamic equation:(1)xΔt+ptxσt=rt+qtut,t∈0,TT.At the same time, in order to keep steady development, we may assume that the number of sheep at the beginning is equal to that at the end, that is,(2)x0=xσT.Suppose thatxu is the solution of the controlled systems (1) and (2) corresponding to the control policy u and xd is the desired value. Recently, the authors [19] considered the optimal control problem P0. Find a u0∈Uad such that(3)Ju0≤Ju,for allu∈Uad,where(4)Ju=∫0Txuσt−xdt2Δt+∫0Tu2tΔt,u∈Uad,is the quadratic cost functional.Motivated greatly by the abovementioned works, in this paper, we suppose that the controlled system is governed by the following more general nonlinear periodic boundary value problem:(5)xΔt+ptxσt=ft,xt,xσt+gut,t∈0,TT,x0=xσT.First, by imposing some suitable conditions onp,f, and g, for any given control policy u∈Uad, we obtain the existence of a unique solution xu for the nonlinear controlled system (5). Then, we study the optimal control problem P. Find a u0∈Uad such that(6)Ju0≤Ju,for allu∈Uad,where(7)Ju=∫0Txuσt−xdt2Δt+∫0ThutΔt,u∈Uad,where xd is the desired value and h:ℝ⟶0,∞ is continuous.


## 2. Preliminaries

In this section, we will provide some foundational definitions and results from the calculus on time scales.Definition 1.
We define the forward jump operatorσ:T⟶T by(8)σt:=infs∈T:s>t,for allt∈T,while the backward jump operator ρ:T⟶T is defined by(9)ρt:=sups∈T:s<t,for allt∈T.
In this definition, we putinf∅=supT and sup∅=infT, where ∅ denotes the empty set. If σt>t, we say that t is right-scattered, while if ρt<t, we say that t is left-scattered. Also, if t<supT and σt=t, then t is called right-dense, and if t>infT and ρt=t, then t is called left-dense. If T has a left-scattered maximum m, then we define Tk=T−m, otherwise Tk=T. Finally, the graininess function μ:T⟶0,∞ is defined by(10)μt:=σt−t,for allt∈T.Definition 2.
Assumef:T⟶ℝ is a function and let t∈Tk. Then, we define fΔt to be the number (provided it exists) with the property that given any ε>0, there is a neighborhood U of ti.e.,U=t−δ,t+δTfor someδ>0 such that(11)fσt−fs−fΔtσt−s≤εσt−s,for alls∈U.
We callfΔt the delta derivative of f at t.
Moreover, we say thatf is delta differentiable (or in short, differentiable) on Tk provided fΔt exists for all t∈Tk. The function fΔ:Tk⟶ℝ is then called the (delta) derivative of f on Tk. A function F:T⟶ℝ is called an antiderivative of f:T⟶ℝ provided(12)FΔt=ftholds for allt∈Tk.
IfF:T⟶ℝ is an antiderivative of f:T⟶ℝ, then we define the Cauchy integral by(13)∫abftΔt=Fb−Fa,for alla,b∈T.Definition 3.
A functionf:T⟶ℝ is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T.Definition 4.
We say that a functionp:T⟶ℝ is regressive provided(14)1+μtpt≠0,for allt∈Tk,holds. The set of all regressive and rd-continuous functions will be denoted by ℛ. We define the set of positively regressive functions ℛ+ as the set consisting of those p∈ℛ satisfying(15)1+μtpt>0,for allt∈T.Lemma 1.
Letp∈ℛ, t0∈T, and ep⋅,t0 be the exponential function on T. Then,(i)
ept,t≡1for allt∈T(ii)
epΔt,t0=ptept,t0for allt∈Tk
Moreover, ifp∈ℛ+, then(16)ept,t0>0,for allt∈T.Lemma 2.
Letf be a continuous function on a,bT that is differentiable on a,bT. Then, f is increasing, decreasing, nondecreasing, and nonincreasing on a,bT if fΔt>0, fΔt<0, fΔt≥0, and fΔt≤0 for all t∈a,bT, respectively.Lemma 3.
Letfnn=1∞ be a sequence of Δ integrable functions on a,bT and suppose that fn⟶f uniformly on a,bT for a function f defined on a,bT. Then, f is Δ integrable from a to b and(17)∫abftΔt=limn⟶∞∫abfntΔt.In the remainder of this paper, we always assume that Banach space(18)Ca,bT,ℝ=yy:a,bT⟶ℝis continuous,is equipped with the norm y=maxt∈a,bTyt, p:0,TT⟶0,∞ is rd-continuous, and denote(19)M=1epσT,0−1.Then, it is easy to see thatM>0.Lemma 4 (see [20]).
For anyy∈C0,TT,ℝ, the following first order linear periodic boundary value problem(20)xΔt+ptxσt=yt,t∈0,TT,x0=xσT,has a unique solution(21)xt=1ept,0∫0teps,0ysΔs+M∫0σTeps,0ysΔs,t∈0,σTT.


## 3. Main Results

First, we list the following two conditions which we shall use in the sequel.A1f:0,TT×ℝ2⟶ℝ is continuous and there exists 0<L<M/21+M2σT such that(22)ft,ω1,v1−ft,ω2,v2≤Lω1−ω2+v1−v2,t∈0,TT,ω1,ω2,v1,v2∈ℝ.A2g:ℝ⟶ℝ and there exists K>0 such that(23)gω−gv≤Kω−v,ω,v∈ℝ.From now on, we always suppose that the control space isC0,TT,ℝ and the admissible control set Uad is a compact subset of C0,TT,ℝ.Lemma 5.
Assume that conditionsA1 and A2 are satisfied. Then, for any given control policy u∈Uad, the nonlinear controlled system (5) has a unique solution xu and(24)xut=1ept,0∫0teps,0fs,xus,xuσs+gusΔs+M∫0σTeps,0fs,xus,xuσs+gusΔs,t∈0,σTT.Proof.
For any fixedu∈Uad, we define an operator Φu:C0,σTT,ℝ⟶C0,σTT,ℝ as follows:(25)Φuxt=1ept,0∫0teps,0fs,xs,xσs+gusΔs+M∫0σTeps,0fs,xs,xσs+gusΔs,t∈0,σTT.
Obviously,x is a solution of the nonlinear controlled system (5) if and only if x is a fixed point of Φu in C0,σTT,ℝ.
Letx,y∈C0,σTT,ℝ. Then, in view of Lemmas 1 and 2 and A1, we have(26)Φuxt−Φuyt=1ept,0∫0teps,0fs,xs,xσs−fs,ys,yσsΔs+M∫0σTeps,0fs,xs,xσs−fs,ys,yσsΔs≤1ept,0∫0teps,0fs,xs,xσs−fs,ys,yσsΔs+M∫0σTeps,0fs,xs,xσs−fs,ys,yσsΔs≤1+M∫0σTeps,0fs,xs,xσs−fs,ys,yσsΔs≤L1+M∫0σTeps,0xs−ys+xσs−yσsΔs≤2L1+Mx−y∫0σTeps,0Δs≤2L1+M2σTMx−y,t∈0,σTT,so(27)Φux−Φuy≤2L1+M2σTMx−y,which together with 0<L<M/21+M2σT implies that Φu:C0,σTT,ℝ⟶C0,σTT,ℝ is a contraction mapping.
Therefore, it follows from Banach contraction principle thatΦu has a unique fixed point xu∈C0,σTT,ℝ. This indicates that the nonlinear controlled system (5) has a unique solution xu and(28)xut=1ept,0∫0teps,0fs,xus,xuσs+gusΔs+M∫0σTeps,0fs,xus,xuσs+gusΔs,t∈0,σTT.Theorem 1.
Assume that conditionsA1 and A2 are satisfied and h:ℝ⟶0,∞ is continuous. Then, the optimal control problem P has an optimal solution u0∈Uad.Proof.
First, it follows from Lemma5 that, for any given control policy u∈Uad, the nonlinear controlled system (5) has a unique solution xu and(29)xut=1ept,0∫0teps,0fs,xus,xuσs+gusΔs+M∫0σTeps,0fs,xus,xuσs+gusΔs,t∈0,σTT.
Next, in view ofJu=∫0Txuσt−xdt2Δt+∫0ThutΔt,u∈Uad, it is obvious that infu∈UadJu exists. Thus, by the definition of infimum, we know that there exists a sequence unn=1∞⊂Uad such that(30)limn⟶∞Jun=infu∈UadJu.
On the one hand, sinceUad is a compact subset of C0,TT,ℝ and unn=1∞⊂Uad, unn=1∞ has a convergent subsequence in Uad. Without loss of generality, we may assume that unn=1∞ converges in Uad, that is, there exists u0∈Uad such that(31)limn⟶∞un=u0.
On the other hand, in view of Lemmas1 and 2, A1 and A2, for any n=1,2,…, we have(32)xunt−xu0t=1ept,0∫0teps,0fs,xuns,xunσs−fs,xu0s,xu0σs+guns−gu0sΔs+M∫0σTeps,0fs,xuns,xunσs−fs,xu0s,xu0σs+guns−gu0sΔs≤∫0teps,0fs,xuns,xunσs−fs,xu0s,xu0σs+guns−gu0sΔs+M∫0σTeps,0fs,xuns,xunσs−fs,xu0s,xu0σs+guns−gu0sΔs≤1+M∫0σTeps,0fs,xuns,xunσs−fs,xu0s,xu0σs+guns−gu0sΔs≤1+M∫0σTeps,0fs,xuns,xunσs−fs,xu0s,xu0σs+guns−gu0sΔs≤1+M∫0σTeps,0Lxuns−xu0s+xunσs−xu0σs+Kuns−u0sΔs≤1+M2Lxun−xu0+Kun−u0∫0σTeps,0Δs≤2L1+M2σTMxun−xu0+K1+M2σTMun−u0,t∈0,σTT,so, for any n=1,2,…, we obtain(33)xun−xu0≤K1+M2σTM−2L1+M2σTun−u0,which together with (31) implies that(34)limn⟶∞xun=xu0.
Thus, in view of Lemma3, (31), and (34), we obtain(35)limn⟶∞Jun=limn⟶∞∫0Txunσt−xdt2Δt+∫0ThuntΔt=∫0Tlimn⟶∞xunσt−xdt2Δt+∫0Tlimn⟶∞huntΔt=∫0Txu0σt−xdt2Δt+∫0Thu0tΔt=Ju0,which together with (30) indicates that(36)Ju0=infu∈UadJu.
Therefore,Ju0≤Ju for all u∈Uad. This shows that u0 is an optimal solution of the optimal control problem P.Example 1.
LetT=0,1∪2,3. We suppose that the controlled system is governed by the following nonlinear periodic boundary value problem(37)xΔt+xσt=Dt2xtarctanxt−12ln1+x2t+π4sin2xσt+ut,t∈0,3T,x0=x3,where(38)D=22e2−19π28e4−4e2+1.
In view ofT=0,1∪2,3, T=3 and pt≡1 for t∈0,3T, it is not difficult to obtain that(39)M=12e2−1.
Sinceft,ω,v=Dt2ωarctanω−1/2ln1+ω2+π/4sin2v for t,ω,v∈0,3T×ℝ2 and gω=ω for ω∈ℝ, it is obvious that f:0,3T×ℝ2⟶ℝ is continuous and A2 is satisfied. Moreover, if we choose L=9πD/2, then 0<L<M/21+M2σT and it follows from Lagrange mean value theorem that(40)ft,ω1,v1−ft,ω2,v2=Dt2ω1arctanω1−12ln1+ω12+π4sin2v1−ω2arctanω2+12ln1+ω22−π4sin2v2≤9Dω1arctanω1−12ln1+ω12−ω2arctanω2+12ln1+ω22+π4sin2v1−sin2v2≤9πD2ω1−ω2+v1−v2=Lω1−ω2+v1−v2,t∈0,3T,ω1,ω2,v1,v2∈ℝ.
This shows thatA1 is fulfilled.
For any given constantN>0, let Uad=u∈C0,3T,ℝu0=0andut1−ut2≤Nt1−t2for allt1,t2∈0,3T. Then, it is easy to verify that Uad is a compact subset of C0,3T,ℝ.
By Lemma5, we know that, for any given control policy u∈Uad, the nonlinear controlled system (37) has a unique solution xu.
Now, we consider the optimal control problemP∗. Find a u0∈Uad such that(41)Ju0≤Ju,for allu∈Uad,where(42)Ju=∫03xuσt−xdt2Δt+∫03u2tΔt,u∈Uad,where xd is the desired value.
Sincehω=ω2 for ω∈ℝ, h:ℝ⟶0,∞ is continuous, thus, all the conditions of Theorem 1 are satisfied. Therefore, it follows from Theorem 1 that the optimal control problem P∗ has an optimal solution u0∈Uad.


## 4. Conclusions

In this paper, we consider a class of optimal control problem governed by nonlinear first order dynamic equation on time scales. First, by imposing some suitable conditions on the related functions and applying Banach contraction principle, for any given control policy, we obtain the existence of a unique solution for the nonlinear controlled system. Next, we prove that the optimal control problem has an optimal solution in the admissible control set. Finally, an example is also given to illustrate the main result of this paper.



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