A maximum principle for optimal control system with endpoint constraints

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1 Wang and Liu Journal of Inequalities and Applications 212, 212: R E S E A R C H Open Access A maimum principle for optimal control system with endpoint constraints Weifeng Wang and Bin Liu * * Correspondence: binliu@mail.hust.edu.cn School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 4374, P.R. China Abstract Pontryagin s maimum principle for an optimal control system governed by an ordinary differential equation with endpoint constraints is proved under the assumption that the control domain has no linear structure. We also obtain the variational equation, adjoint equation and Hamilton system for our problem. MSC: 65K1; 34H5; 93C15 Keywords: Pontryagin s maimum principle; optimal control; Hamilton system; transversality condition; linear structure 1 Introduction Optimal control problems have been studied for a long time and have a lot of practical applications in the fields such as physics, biology and economics, etc. Hamilton systems are derived from Pontryagin s maimum principle, which is known as a necessary condition for optimality. Many results have been obtained both for finite and infinite dimensional control systems such as [1 5]. Regarding the state constraint problems, lots of results are also obtained. For eample,readers can refer to [6, 7] and the references therein. To our best knowledge, to derive the necessary conditions of Pontryagin s maimum principle type for optimal control problems, there are two main perturbation methods. When the control domain is conve, we often use the conve perturbation. When the control domain is non-conve and does not have any linear structure, we usually use the spike perturbation. Many relevant results have been obtained; see [3, 6, 8 1] and the references therein. The two methods have their advantages and disadvantages. The conve variational needs the control domain being conve, but in reality it is not always satisfied. And the spike variational needs more regularity for the coefficients and the solutions to the state equations, especially in the stochastic case. In 21, Lou [7] introduced a new method to study the necessary and sufficient conditions of optimal control problems in the absence of linear structure for the deterministic case. The author gave a local linearization of the optimal control problem along the optimal control, and transformed the original problem into a new relaed control problem. Moreover, he proved the equivalence of the two problems in some sense. Being directly inspired by [7], we are also interested in applying this method to an endpoint constraints optimal control system, which is also in the absence of linear structure. Also, Pontryagin s maimum principle is obtained for our problem. The rest of this paper is organized as follows. Section 2 begins with a general formulation of our state constraints optimal control problem and the local linearization of the 212 Wang and Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 2 of 12 problem is given. In Section 3, we give our main result and its proof. Moreover, we obtain the variational equation, adjoint equation and Hamilton system for our optimal control system. 2 Preliminaries We consider the controlled ordinary differential equation in R n ẋt) =f t, t), ut)), ) =, in [, T], 2.1) with the cost functional J, u ) ) = l t, t), ut) ) dt, 2.2) where T >isagivenconstant, R n is a decision variable and u ) U[, T]with U[, T]= { v :[,T] V v )ismeasurable }, where V is a non-conve set in R k. Let U ad [, T] be the set of all elements, u )) S 1 U[, T]satisfying T) S 2, where S 1 and S 2 are closed conve subsets of R n.lets = S 1 S 2. Then the optimal control problem can be stated as follows. Problem 2.1 Find a pair, ū )) U ad [, T]suchthat J, ū ) ) = inf J, u ) ). 2.3),u )) U ad [,T] Any, ū )) U ad [, T] satisfying the above identity is called an optimal control, and the corresponding state ;, ū )) ) is called an optimal trajectory; ), ū )) is called an optimal pair. Let the following hypotheses hold: H 1 ) The metric space V, d) is separable, and d is the usual metric in R k. H 2 ) Functionsf =f 1, f 2,...,f n ) :[,T] R n V R n and l :[,T] R n V R are measurable in t, continuous in, u) and continuously differentiable in, where denotes the transpose of a matri. Moreover, there eists a constant L >such that f t, 1, u) f t, 2, u) L 1 2, lt, 1, u) lt, 2, u) L 1 2, f t,,u) L, lt,,u) L, t, 1, 2, u) [, T] R n R n V, 2.4) where denotes the usual Euclidean norm.

3 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 3 of 12 From the above conditions, it is easy to know that the state equation 2.1)hasaunique solution ;, u )) for any, u )) U[, T]. Let, ū )) U ad [, T] be a minimizer of J, u )) over U ad [, T], and we linearize U ad [, T]along, ū )) for any fied ) in the following manner. Define M ad [, T] { 1 α)δū ) + αδ u ) ) α [, 1],, u ) ) U ad [, T] }, where δ u ) denotes the Dirac measure at u )onu[, T]. Let σ )=1 α)δū ) + αδ u ),then it is easy to see that, σ )) M ad [, T]. Now, we define f t,, σ t) ) f t,, v)σ t) dv =1 α)f t,, ūt) ) + αf t,, ut) ), 2.5) U and similarly, l t,, σ t) ) lt,, v)σ t) dv =1 α)l t,, ūt) ) + αl t,, ut) ). U Then we define )= ;, σ )) as the solution of the following equation: ẋt)=f t, t), σ t)), in [, T], ) =, 2.6) and the corresponding cost functional is J, σ ) ) = l t, t, σ t) ), σ t) ) dt. We can easily find that ;, u )) and J, u )) coincide with ;, δ u ) )andj, δ u ) ) respectively. Thus, U ad [, T] canbeviewedasasubsetofm ad [, T] in the sense of identifying, u )) U ad [, T] and, δ u ) ) M ad [, T]. Because the elements of M ad [, T] are very simple, we need neither pose additional assumptions like that the control domain is compact as Warga [11] did nor introduce the relaed control defined by finite-additive probability measure as Fattorini [12] did. Now, we can see that M ad [, T] already has a linear structure at, ū )).First,wegivesomelemmastoshowthat, δū ) ) is a minimizer of J, σ )) over M ad [, T]. Lemma 2.2 Lou [7]) Let H 1 )-H 2 ) hold. Then there eists a positive constant C >,such that for any, σ )) M ad [, T] and t 1, t 2 [, T], ;, σ )) C[,T] C, 2.7) t 1 ;, σ )) t 2 ;, σ )) C t 1 t 2. Lemma 2.3 Lou [7]) Let H 1 )-H 2 ) hold and, ū )) be a minimizer of J, u )) over U ad [, T]. Then, δū ) ) is a minimizer of J, σ )) over M ad [, T]. Remark 2.4 We can see that Lemma 2.3 shows the equivalence of the above two problems. And this lemma is essential for our following maimum principle.

4 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 4 of Pontryagin s maimum principle In this section, we give our main result first and then prove it. Let f denote the derivative of f on, and others can be defined in the same way., denotes the inner product in R n. Theorem 3.1 Pontryagin s maimum principle) We assume H 1 )-H 2 ) hold. Let ), ū )) be a solution of the optimal control problem 2.1). Then there eists a nontrivial pair ψ, p )) R C 1 [, T]; R n ), i.e.,ψ, p )) such that ψ, 3.1) [ ) pt)=pt)+ f s, s), ūs) ps) t + ψ l s, s), ūs) )] ds, a.e. t [, T], 3.2) p), 1 pt), 2 T), 1, 2 ) S, 3.3) H t, t), ūt), pt), ψ ) = ma,u )) U ad [,T] H t, t), ut), pt), ψ ), a.e. t [, T], 3.4) where for any t,, u, p, ψ ) [, T] R n V R n R, H t,, u, p, ψ ) = f t,, u), p + ψ lt,, u), 3.5) and we also have the following Hamilton system: ẋt)=h p t, t), ut), pt), ψ ), a.e. t [, T], ṗt)= H t, t), ut), pt), ψ ), a.e. t [, T], ) =, pt)= ψ, Ht, t), ut), pt), ψ ) = ma,u )) U ad [,T] Ht, t), ut), pt), ψ ), a.e. t [, T], 3.6) where ψ will be defined in the following part. We recall that under the conditions H 1 )-H 2 ), for any, u )) R n U[, T], the state equation 2.1) admits a unique solution ) with) =.So,thecostfunctional is uniquely determined by, u )). In the sequel, we denote the unique solution of 2.1) with ) = by ;, u )). Now, let, ū )) U ad [, T] be an optimal control. We denote ) =. Without loss of generality, we may assume that J, ū )) = ; otherwise, we may consider the optimal control problem with a cost functional of J, u )) J, ū )). Now, we give some definitions and lemmas for Theorem 3.1. First, we define a penalty functional, via which, for convenience, we can transform the original problem to another one called the approimate problem, which has no endpoint constraint. Let us introduce some notations. For all 1, u 1 )), 2, u 2 )) R n U[, T], we define d 1, u 1 ) ), 2, u 2 ) )) = d u 1 ), u 2 ) ), 3.7)

5 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 5 of 12 where du 1 ), u 2 )) denote the measure of {t [, T] u 1 t) u 2 t)} it obviously is a metric). For >and, u )) R n U[, T], the penalty functional is defined as follows: J, u ) ) = { d 2 S, T;, u ) )) + [ J, u ) ) + ) +] 2 } 1 2, 3.8) where a) + = ma{, a},andforany, 1 ) R n R n, d S, 1 )=d, 1 ), S ) inf y,y 1 ) S { y 2 + y 1 1 2} ) Obviously, d S is a conve function and it is Lipschitz continuous with the Lipschitz constant being 1. We define the subdifferential of the function d S as follows: d S, 1 )= { a, b) R n R n d S, 1) ds, 1 ) a, + b, 1 1,, 1) R n R n}, 3.1) where, denotes the inner product in R n. For more properties of subdifferential d S, one can see p.146 in [6]. Lemma 3.2 Let H 1 )-H 2 ) hold. Then there eists a constant C >such that for all, u )), ˆ, û )) R n U[, T], sup t [,T] t;, u )) t; ˆ, û )) C1 + ˆ ) d, u )), ˆ, û ))), J, u )) Jˆ, û )) C1 + ˆ ) d, u )), ˆ, û ))). 3.11) Proof Denote )=t;, u )) and ˆ )=t; ˆ, û )).Fromthestateequation2.1) and condition H 2 ), we have t) ) f s, s), us) ds f s,,us) ) + f s, s), us) ) f s,,us) ) ) ds L + L s) ) ds. By Gronwall s inequality, it follows that t) C 1+ ), t [, T]. Similarly, ˆt) C 1+ ˆ ), t [, T],

6 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 6 of 12 where the constant C is independent of controls u ) and û ), and may be different at different places throughout this paper. Further, noting the definition of d, we have t) ˆt) C ˆ + C + s) ˆs) ds f s, s), us) ) f s, s), ûs) ) ds C ˆ + C 1+ ˆ ) d u ), û ) ) + C C 1+ ˆ ) d, u ) ), ˆ, û ) )) + C Thus, by Gronwall s inequality, we get t) ˆt) C 1+ ˆ ) d, u ) ), ˆ, û ) )) e Ct s) ˆs) ds s) ˆs) ds. C 1+ ˆ ) d, u ) ), ˆ, û ) )) e CT, t [, T]. Taking supremum in the above inequality, the first inequality in 3.11) isobtained. The second inequality can be proved similarly. By the definition of J, u )) and Lemma 3.2, we can easily obtain the following result. Corollary 3.3 The functional J, u )) is continuous on the space R n U[, T], d). Remark 3.4 By the definition of J, u )) 3.8)andCorollary3.3,wecanseethat J, u )) >,, u )) R n U[, T], 3.12) J, ū )) = inf R n U[,T] J, u )) +. Thus, by the Ekeland variational principle see p.135 in [6] for details), there eists a pair, u )) R n U[, T], such that d, u ) ),, ū ) )), 3.13) J, u ) ) + d, u ) ),, u ) )) J, u ) ),, u ) ) R n U[, T]. 3.14) The above implies that if we let )= ;, u )), then ), u )) is an optimal pair for the problem where the state equation is 2.1) and the cost functional is J, u )). Now, we derive the necessary conditions for ), u )). Lemma 3.5 Let ), u )) be an optimal pair for the problem where the state equation is 2.1) and the cost functional is J, u )), then there eists a nontrivial triple ψ, ϕ, ψ ) R R n R n such that ϕ 2 + ψ 2 + ψ 2 =1,

7 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 7 of 12 and η + C ) ϕ, η + ψ, X T) + ψ Y, where η, X ), Y will be defined in the following proof. Proof Similar to Section 2, welinearizeu[, T] alongu ) and denote σ α, ) =1 α)δ u ) + αδ u ), α, )= ; + αη, σ α, )), where η B 1 ) R n B 1 ) denotes the ball whosecenterisandradiusis1).recallthat )= ;, u )), then we derive the variational equation. From 2.5), we have X α, t) α, t) t) α [ f s, α, s), u s)) f s, s), u s)) = α = + f s, α, s), us) ) f s, α, s), u s) )] ds [ 1 f s, s)+τ α, s) s) ), u s) ) dτx α, s) + f s, α, s), us) ) f s, α, s), u s) )] ds, where f t,, u) denotes the transpose of the Jacobi matri of f on.byvirtueofh 2 ), and using the convergence of α, t) t)seetheproofoflemma2.2in[7]), we can easily obtain X α, t) X t), as α, t [, T], where X t) is the solution of the following variational equation: Ẋ t)=f t, t), u t)) X t)+ft, t), ut)) f t, t), u t)), X ) = η. 3.15) From the definition of σ α, ), by Lemma 2.3, for any fied α,wecandefine d σ α, ), δ u )) = αd u ), u ) ), and d 1, σ α, ) ), 2, δ u )) ) = d σ α, ), δ u )). So, by virtue of 3.14) and Lemma 2.3,wehave J + αη, σ α, ) ) + d + αη, σ α, ) ),, δ u ))) J, δ u )). 3.16)

8 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 8 of 12 It means that when α, η + d u ), u ) )) 1 lim J α α + αη, σ α, ) ) J, δ u ))) 1 = 2J, δ u )) lim 1 { d 2 α α S + αη, α, T) ) ds 2, T) ) + [ J + αη, σ α, ) ) + ) +] 2 [ J, δ u )) + ) + ] 2 }. Note that the map, 1 ) d 2 S, 1 ) is continuously differentiable on R n R n,thenwe get lim α 1 { d 2 α S + αη, α, T) ) ds 2, T) )} =2d S, T) ) a, η + b, X T) ), 3.17) with a, b ) d S, T)) and a 2 + b 2 = ) Similarly, we can obtain lim α 1 {[ J α + αη, σ α, ) ) + ) +] 2 [ ) J + ] 2 }, δ u )) + =2 J, δ u )) + ) +Y, 3.19) where J1 α)δ u Y = lim ) + αδ u ) ) Jδ u )) α α [ lt, α, t), u t)) lt, t), u t)) = lim α α + l t, α, t), ut) ) l t, α, t), u t) )] dt [ 1 = lim l t, t)+s α, t) t) ), u t) ), X α, t) ds α = + l t, α, t), ut) ) l t, α, t), u t) )] dt [ l t, t), u t) ), X t) + l t, t), ut) ) l t, t), u t) )] dt. Combining 3.17) and3.19), by sending α, we obtain η + C ) ϕ, η + ψ, X T) + ψ Y, 3.2) where ϕ, ψ )= d S, T)) J,δ u ) ) a, b ), J ψ =,δ u ) )+)+ J,δ u ) ). 3.21)

9 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 9 of 12 From 3.18), it is obvious that ϕ 2 + ψ 2 + ψ 2 = ) Remark 3.6 By the definition of subdifferential of the function d S ), for any 1, 2 ) S, we have ϕ, 1 + ψ, 2 T). 3.23) Inordertopasstothelimitas, we give the following lemma first, which is necessary for the derivation of our maimum principle. Lemma 3.7 It holds that [ lim + sup X t) Xt) ] + Y Y =, t [,T] where Xt) and Y satisfy the following equations: Ẋt)=f t, t), ūt)) Xt)+f t, t), ut)) f t, t), ūt)), X) = η, 3.24) Y = [ l t, t), ūt) ) Xt)+l t, t), ut) ) l t, t), ūt) )] dt. 3.25) Proof By 3.13) and the definition of d,itiseasytoseethat lim =. 3.26) From 3.15)and3.24), we have X t) Xt) f s, s), u s) ) X s)+f s, s), us) ) f s, s), u s) ) f s, s), ūs) ) Xs)+f s, s), us) ) f s, s), ūs) )) ds [ f s, s), u s) ) X s) Xs) + f s, s), u s) ) ) f s, s), ūs) Xs) + f s, s), us) ) f s, s), us) ) + f s, s), u s) ) f s, s), ūs) ) ] ds. 3.27) By virtue of Lemma 3.2,then sup t) t) t [,T] C 1+ ) d, u ) ),, ū ) )), as. 3.28)

10 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 1 of 12 Now, combining the condition H 2 )and3.27), we deduce that X t) Xt) C 1+ ) d, u ) ),, ū ) )) + C Then by Gronwall s inequality, we obtain lim sup t [,T] X s) Xs) ds. 3.29) X t) Xt) =. 3.3) Similarly, we can prove that lim Y Y =. 3.31) Thus, the proof of this lemma is completed. Remark 3.8 Now, we can let. By 3.23), it is obvious that for any 1, 2 ) S, we have ϕ, 1 + ψ, 2 T) 2 + T) T) 2 ) 1 2 δ. 3.32) Hence, combining 3.2)and3.32), for any 1, 2 ) S,weobtain ϕ, η 1 ) + ψ, XT) 2 T) ) + ψ Y η + C ) δ X T) XT) Y Y. 3.33) From 3.22), we can find a subsequence still denoted by itself) such that ϕ, ψ, ψ ) ϕ, ψ, ψ ). 3.34) Now, by Lemma 3.7 and sending in3.33), for any 1, 2 ) S, u ) U[, T] and η B 1 ), we have ϕ, η 1 ) + ψ, XT) 2 T) ) + ψ Y. 3.35) Based on the above preparation, now we start to prove Theorem 3.1 by the duality relations. Proof of Theorem 3.1 Let pt) solve the adjoint equation ṗt)= f t, t), ūt))pt) ψ l t, t), ūt)), pt)= ψ, 3.36) where ψ = ψ.

11 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 11 of 12 So, by virtue of 3.35), we get ϕ, 1 η + pt), XT) 2 T) ) + ψ Y. 3.37) Simultaneously, we have the following duality equality: pt), XT) p), η + ψ Y ) = f t, t), ūt) pt) ψ ) l t, t), ūt), Xt) = = + ) Xt)+f ) ) pt), f t, t), ūt) t, t), ut) f t, t), ūt) dt + ψ [ ) Xt)+l ) )] l t, t), ūt) t, t), ut) l t, t), ūt) dt pt), f t, t), ut) ) f t, t), ūt) ) + ψ l t, t), ut) ) l t, t), ūt) )) dt [ H t, t), ut), pt), ψ ) H t, t), ūt), pt), ψ )] dt, 3.38) where H is the Hamilton function defined in 3.5). Now, setting η =and 1, 2 )=, T)) in 3.37), we obtain [ H t, t), ut), pt), ψ ) H t, t), ūt), pt), ψ )] dt. As U is separable and, u )) U ad [, T] is arbitrary, the above inequality can be written as H t, t), ut), pt), ψ ) H t, t), ūt), pt), ψ ), a.e. t [, T]. 3.39) Thus, H t, t), ūt), pt), ψ ) = ma,u )) U ad [,T] H t, t), ut), pt), ψ ), a.e.t [, T]. 3.4) Net, by taking ut) =ūt) and 1, 2 )=, T)) in 3.37), using the duality equality 3.38), we get ϕ, η pt), XT) + ψ Y = p), η, η B 1 ). 3.41) Thus, ϕ = p). Then taking η = and ut) = ūt) in3.37), combining with the duality equality 3.38), we obtain the transversality condition p), 1 pt), 2 T), 1, 2 ) S. 3.42) Finally, we claim that ψ, p )). Otherwise, in particular, we have that ψ = pt)=, ϕ = p) =. 3.43)

12 Wang and Liu Journal of Inequalities and Applications 212, 212: Page 12 of 12 Note that ψ = in this case, so this gives a contradiction to 3.22). The Hamilton system 3.6) is obvious. Then the proof of the maimum principle is completed. Remark 3.9 We give some important special cases of our control problem. i) The control problem with fied endpoints. In this case, the constraint set is S = {, T )} and the endpoint constraint becomes of the following form: ) =, T)= T. ii) The control problem with a terminal state constraint, i.e., ) =, T) S 2. Competing interests The authors declare that they have no competing interests. Authors contributions All authors considered this problem and carried out the proof. We all also read and approved the final manuscript. Acknowledgements We would like to thank the referees for the useful suggestions which helped to improve the previous version of this paper. This work was partially supported by NNSF of China Grant No ). Received: 3 May 212 Accepted: 28 September 212 Published: 12 October 212 References 1. Barbu, V: Optimal of Variational Inequalities. Pitman, London 1984) 2. Barbu, V, Da Prato, G: Hamilton Jacobi Equations in Hilbert Spaces. Pitman, London 1983) 3. Barbu, V, Precupanu, T: Conveity and Optimization in Banach Spaces. Reidel, Dordrecht 1986) 4. Capuzzo-Dolcetta, I, Evans, LC: Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22, ) 5. Pontryagin, LS: The maimum principle in the theory of control processes. In: Proc. 1st. Congress IFAC, Moscow 196) 6. Li, XJ, Yong, YM: Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston 1995) 7. Lou, HW: Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete Contin. Dyn. Syst., Ser. B, 14, ) 8. Casas, E: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24, ) 9. Casas, E, Fernández, LA: Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Appl. Math. Optim. 27, ) 1. Yong, JM, Zhou, XY: Stochastic Controls: Hamiltonian System and HJB Equations. Springer, New York 1999) 11. Warga, J: Optimal Control of Differential and Functional Equations. Academic Press, New York 1972) 12. Fattorini, HO: Relaed controls in infinite dimensional systems. Int. Ser. Numer. Math. 1, ) doi:1.1186/ x Cite this article as: Wang and Liu: A maimum principle for optimal control system with endpoint constraints. Journal of Inequalities and Applications :.

Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions

$Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions$ Sudsutad and Tariboon Advances in Difference Equations 212, 212:93 http://www.advancesindifferenceequations.com/content/212/1/93 R E S E A R C H Open Access Boundary value problems for fractional differential