# Stability Analysis of Optimal Trajectory for Nonlinear Optimal Control Problems

Authors: Hongyong Deng; Wei Zhang; Changchun Shen
Journal: Journal of Mathematics
 (2020)

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## Abstract

Due to the need for numerical calculation and mathematical modelling, this paper focuses on the stability of optimal trajectories for optimal control problems. The basic ideas and techniques are based on the compactness of the optimal trajectory set and set-valued mapping theorem. Through lack of optimal control stability, the result of generic stability for optimal trajectories is obtained under the perturbations of the right-hand side functions of the state equations; in the sense of Baire category, the right-hand side functions of the state equations of optimal control can be approximated by other functions.


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## Body

## 1. Introduction

Due to the need for numerical calculation and mathematical modelling, we will consider the influence of optimal trajectory with the changing right-hand side functions of the state equations on the approximate functions. There are many experts who have done much work on the stability of the optimal control problem [1–4], and especially in the case of the disturbance of the right-hand side function, the stability of the optimal control is discussed [5–7]. However, in the actual problem, the real decisive factor for the optimal control problem is the stability of the optimal trajectory, and sometimes, the optimal control does not necessarily converge; that is, when the trajectory converges, the corresponding control does not necessarily converge. Therefore, this paper discusses the stability in these cases, focusing on two problems: First, the compactness of the feasible trajectories corresponding to the perturbations of right-hand side functions of the state equations is discussed, and then, the compactness of the optimal trajectories is discussed. Second, the stability of the optimal trajectories set corresponding to the perturbations of the right-hand side functions is discussed.Consider Bolza problem, Problem P, as follows:Find an optimal pairy¯⋅,u¯⋅∈pad0,T such that the cost functional(1)Jy⋅,u⋅=hyT+∫0Tf0t,ytdtreaches its minimum at y¯⋅,u¯⋅for ally⋅,u⋅∈Pad0,T, where the admissible pair y⋅,u⋅ are solutions of the following nonlinear controlled system:(2)y˙t=ft,yt,ut,t∈0,T,y0=y0.Here,ut∈U0,T and Pad0,T is the set of all admissible pair, and the set of admissible controls is defined by(3)U0,T≡u:0,T⟼Uu⋅is measurable,where U is a metric space.In order to discuss the optimal control problem (P), let us start with some basic assumptions:(H1) The terminal timeT>0 is fixed, and the metric space U is compact.(H2) The functionf:0,T×Rn×U⟼Rn is measurable w.r.t. t∈0,T, continuous concerning u∈U. There exist L,C>0 such that(4)ft,x,u−ft,y,u≤Lx−y,ft,x,u≤C,for allx,y∈Rn,t∈0,T,u∈U.(H3) The functionf:0,T×Rn×U⟼Rn satisfies the following Filippov–Roxin condition:The setft,y,U=ft,y,uu∈U is convex and closed for almost all t∈0,T and all y∈Rn(H4) The functionf0:0,T×Rn⟼R is Borel measurable concerning t∈0,T and continuous concerning y∈Rn, and h:Rn⟼R is continuous.Under the assumptions and from [8], equation (2) has a unique solution. Also, the solution of equation (2) depends continuously on the right-hand side function and is differentiable concerning initial data.This paper is organized as follows. In part one, we construct the complete metric space off satisfied the conditions of (H2-H3) and discuss the compactness of the feasible trajectories set. In part two, we consider the stability of optimal trajectory in sense of Baire category. In the last section, we give an example and conclusion.


## 2. Compactness of Optimal Trajectory

For the aim to consider the stability of optimal trajectory, first, we need some notions.Let(5)Y=ffsatisfies the conditions ofH2andH3.For eachf1,f2∈Y, define(6)ρf1,f2=supt,y,u∈0,T×Rn×Uf1t,y,u−f2t,y,u,then we know that space Y,ρ is a complete metric space easily.Set(7)Sf=y¯y¯are optimal trajectories of optimal control problemsPfor eachf∈Y.Then, the correspondence f⟼Sf is a set-valued mapping; for convenience to discuss with the next, we expand the scope of Sf into YS0,T⊆C0,T;Rn, where YS0,T⊆C0,T;Rn is the set of all feasible trajectories of optimal control problems (P) for each f∈Y, and we denote the set-valued mapping by S:Y⟼2YS0,T.Now, we consider the properties of feasible trajectories setYS0,T⊆C0,T;Rn; we have the following theorem.Theorem 1.
Suppose assumptions (H1) and (H2) hold, then the feasible trajectories setYS0,T is relatively compact in C0,T;Rn.Proof.
By the assumptions of (H1) and (H2), equation (2) has a unique solution for each f∈Y, and the solution is denoted by yft,u⋅=y0+∫0tfτ,yτ,uτdτ. Thanks to the assumption (H2), it follows that(8)yf=y0+∫0tfτ,yτ,uτdτ≤y0+∫0tfτ,yτ,uτdτ≤y0+∫0tCdτ≤y0+∫0TCdτ.
Namely,(9)yft,u⋅≤CT+y0,for eachf∈Y,u⋅∈U0,T.
This means that the feasible trajectories setYS0,T is uniformly bounded.
Furthermore, we obtain(10)yft,u⋅−yfs,u⋅≤∫stfτ,yτ,uτdτ≤∫stCdτ.
Hence,(11)yft,u⋅−yfs,u⋅≤Ct−s,for alls,t∈0,T.
That is, the feasible trajectories setYS0,T is equicontinuous.
From (7)–(11), by the Arzela–Ascoli theorem, we prove this theorem.Theorem 2.
If assumptions (H1), (H2), (H3), and (H4) are satisfied, the feasible trajectories setYS0,T is compact in C0,T;Rn.Proof.
Letyk⋅⊆YS0,T and yk⋅ be the solution of the following equations:(12)yk.t=fkt,ykt,ukt,t∈0,T,yk0=y0,under the assumptions (H1)–(H4) and fk∈Y. Let yk⋅=yfk⋅,y0,uk⋅ and assume that(13)yk⋅⟶y¯⋅inC0,T;Rn.Next, we need to show that y¯⋅∈YS0,T.
Assume that(14)fk⋅,yk⋅,uk⋅⟶wf¯⋅inLp0,T;Rn,where p>1, and we have f¯∈Y. By the Mazur theorem, there exist convex combinations with αik≥0,∑i≥1αik=1, such that(15)∑i≥1αikfk+i⋅,yk+i⋅,uk+i⋅⟶f¯⋅inLp0,T;Rn.By (H2) and (13), we have(16)∑i≥1αikf¯t,yk+it,uk+it−∑i≥1αikf¯t,y¯t,uk+it≤L∑i≥1αikyk+it−y¯t⟶0k⟶+∞,y¯t=limk⟶∞ykt=limk⟶∞y0+∫0tfks,yks,uksds=y0+∫0tf¯sds=y0+∫0tlimk⟶∞∑i≥1αikfk+is,yk+is,uk+isds=y0+∫0tlimk⟶∞∑i≥1αikf¯s,yk+is,uk+isds=y0+∫0tlimk⟶∞∑i≥1αikf¯s,y¯s,uk+isds.
Consider(17)f¯t=limk⟶∞fk=limk⟶∞∑i≥1αikf¯t,y¯t,uk+it∈co¯f¯t,y¯t,U=f¯t,y¯t,U.
By the Filippov’s Lemma, there existsu¯∈U0,T such that (17) is satisfied, namely,(18)y¯=y0+∫0tf¯s,y¯s,u¯sholds.


## 3. Stability of Optimal Trajectory

In this section, we discuss the stability of optimal trajectory based Baire category.Theorem 3.
Let (H1)–(H4) hold andSf≠∅ for each f∈Y.Proof.
For any fixedf∈Y, let yk⋅,uk⋅∈Pad0,T, and we have yk⋅∈YS0,T; by the compactness of YS0,T, there exists y¯⋅=y⋅,y0,u¯⋅∈YS0,Tandu¯⋅∈U0,T, such that yk⋅⟼y¯⋅.
By assumption of (H4), we have(19)f0t,y¯t=limk⟶∞f0t,ykt,hy¯T=limk⟶∞hykT.
Namely,Jfyk⋅⟶Jfy¯⋅, so Jf⋅ is continuous concerning y∈YS0,T; since YS0,T is a compact set, the minimum value of Jf⋅ exists. Then, Sf≠∅ for each f∈Y.Theorem 4.
Let (H1)–(H4) hold, andSf is closed for each f∈Y.Proof.
By the compactness ofYS0,T, next we show that GraphS=f,y∈Y×YS0,Ty∈Sf is closed. Take fk⊂Y with fk⟶f∈Y and y¯k∈Sfk with y¯k⟶y¯∈YS0,T; first, we need to show y¯∈Sf.
Due toy¯k∈Sfk, for all u∈U0,T, we have(20)Jfky¯k≤Jfkyk,and by (H4), we see that(21)Jfky¯k⟶Jfy¯.
For any fixedu∈U0,T in controlled system, we obtain(22)Jfkyk⟶Jfy.
From the abovementioned equations, for allu∈U0,T, the inequality(23)Jfy¯≤Jfyholds. Namely, y¯∈Sf and Sf is closed.
SinceSf⊂YS0,T is closed and YS0,T is a compact set, we have the following theorem.Theorem 5.
Let (H1)–(H4) hold, andSf is compact for each f∈Y.Theorem 6.
Let (H1)–(H4) hold, and then,S is upper semicontinuous at each f∈Y.Proof.
SupposeS is not upper semicontinuous; then, there exists open sets O⊂C0,T;Rn, Sf⊂O, and fk⊂Yandfk⟶f, such that some y¯fk∈Sfk, but y¯fk∉O; in fact, by compactness, there exists a subsequence y¯fk′ of y¯fk such that y¯fk′⟶y¯. Since O is an open set and y¯fk∉O, then y¯fk′∉O; we have that y¯∉O, but y¯∈Sf, and this contradicts y¯∈O. We proved this theorem.
Hence, the set-valued mappingS:Y⟼2YS0,T is compact upper semicontinuous, from the Lemma 4.2 of [6], Definition 3.1 of [5], and Theorem 4.7 of [7], and therefore, we can obtain the theorem of stability.Theorem 7.
There exists a dense residual subsetW of Y, such that for any f∈W, Sf is stable in the sense of Hausdorff metric.Remark 1.
By Theorem7, if f∈Y, the optimal control problem can be approximated in sense of Baire category.Remark 2.
We consider that the cost functionalJy⋅,u⋅=hyT+∫0Tf0t,ytdt contains u⋅ implicitly. Namely, the cost functional is Jy⋅,u⋅=hyT+∫0Tf0t,yt,utdt, and we need to let(24)ft,yt,ut=ft,yt,utf0t,yt,ut,yt=yty0t,y0=y00.Then, we can transform cost functional into Jy⋅,u⋅=hyT+y0T, and the discussion of stability is similar to the problem (P).


## 4. Example

Consider the optimal problem(25)minu∈U0,1Ju=y12subject to the system(26)y˙t=ft,yt,ut,t∈0,1,y0=0,where u∈U0,1 and(27)U0,1=ut∈−1,1,where(28)ut=α,ift∈0,12,β,otherwise,and α,β∈−1,1.One can easily show thatU0,1 satisfies hypothesis H3.(1)
Letft,yt,ut=ut,andt∈0,1. Then, for any α∈−1,1,(29)uαt=α,ift∈0,12,−α,otherwise.We haveJuα=0. That is,(30)Sf=argminu∈U0,1Jfu=yα,α∈−1,1,where the optimal trajectory(31)yαt=αt,ift∈0,12,α−αt,otherwise.(2)
Letfmt,yt,ut=ut+1/m,andt∈0,1, for every m=1,2,…; we have Jum=0, where(32)umt=α−1m,ift∈0,12,−α−1m,otherwise,and α−1/m,−α−1/m∈−1,1.Then,(33)Sfm=argminu∈U0,1Jfmu=ym,where the optimal trajectory(34)ymt=αt,ift∈0,12,α−αt,otherwise.Obviously,ρf,fm=1/m→0 as m→+∞ and ym→yα.


## 5. Conclusions

In this paper, under the condition of the lack of good properties of optimal control, we discuss the compactness of the feasible trajectories set when the right-hand side function of the controlled system is disturbed and give a suitable metric for the space of the right-hand side function. Combined with the given assumptions, the space is a complete metric space. Then, according to the compact upper semicontinuous set-valued mapping, the continuous dependence of the optimal trajectory on the right-hand function and the stability of the cost functional on the right-hand function are obtained in the sense of Baire category.



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*Source: 1392705-2020-08-27.xml*