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Journal of Control Science and Engineering
Volume 2011 (2011), Article ID 127208, 9 pages
http://dx.doi.org/10.1155/2011/127208
Research Article

Higher-Order Generalized Invexity in Control Problems

Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, India

Received 13 December 2010; Accepted 30 March 2011

Academic Editor: Onur Toker

Copyright © 2011 S. K. Padhan and C. Nahak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a higher-order duality (Mangasarian type and Mond-Weir type) for the control problem. Under the higher-order generalized invexity assumptions on the functions that compose the primal problems, higher-order duality results (weak duality, strong duality, and converse duality) are derived for these pair of problems. Also, we establish few examples in support of our investigation.

1. Introduction

We consider the control problem(CP)min𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡,(1)subjectto𝑔(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))0,(2)𝐺(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=0,(3)𝑥(𝑎)=𝛾1,𝑥(𝑏)=𝛾2;𝑢(𝑎)=𝛿1,𝑢(𝑏)=𝛿2,(4) where 𝑓, 𝑔, and 𝐺 are twice continuously differentiable functions from 𝐼×𝑛×𝑛×𝑚 into , 𝑟, and 𝑠, respectively, 𝐼=[𝑎,𝑏].

Mangasarian [1] formulated a class of higher-order dual problems for a nonlinear programming problems involving twice continuously differentiable functions. He did not prove the weak duality and hence gave a limited strong duality theorem. Mond and Zhang [2] introduced invexity type conditions under which duality holds between Mangasarian [1] primal problem and various higher-order dual programming problems.

One practical advantage of higher-order duality is that it provides tighter bounds for the value of the objective function of the primal problem when approximations are used, because there are more parameters involved. Higher-order duality in nonlinear programming has been studied by several researchers like Mond and Zhang [2], Chen [3], Mishra and Rueda [4], and Yang et al. [5]. Recently, Gulati and Gupta [6] studied the higher-order symmetric duality over arbitrary cones for Wolfe and Mond-Weir type models. Obtained duality results for various higher-order dual problems under higher-order and type higher-order duality to higher-order type I functions.

Bhatia and Kumar [7] have studied the multiobjective control problems under 𝜌-pseudoinvexity, 𝜌-strictly pseudoinvexity, 𝜌-quasi-invexity, and 𝜌-strictly quasi-invexity assumptions. Nahak and Nanda [8] have studied the efficiency and duality for multiobjective control problems under (𝐹𝜌) convexity. Again Nahak and Nanda [9] proposed a sufficient condition for solutions and duality for the multiobjective variational control problems under 𝑉-invexity. Recently, Padhan and Nahak [10] considered a class of constrained nonlinear control primal problem and formulated the second-order dual. He also gave some duality results (weak duality, strong duality, and converse duality) under generalized invexity assumptions. But in our knowledge, no one has talked about higher-order duality for the control problem. In this paper, we study both Mangasarian and Mond-Weir type higher-order duality of the control primal problem (CP). We give more general type conditions that is higher-order generalized invexity under which duality holds between (CP) and (MHCD), and (CP) and (MWHVD). Our approach is similar to that of Mangasarian [1]. Again, we discuss many counterexamples to justify our work.

2. Notations and Preliminaries

Let 𝑆(𝐼,𝑛) denote the space of piecewise smooth functions 𝑥 with norm 𝑥=𝑥+𝐷𝑥, where the differentiation operator 𝐷 is given by𝑢=𝐷𝑥𝑥(𝑡)=𝜅+𝑡𝑎𝑢(𝑠)𝑑𝑠,(5) where 𝜅 is a given boundary value; thus, 𝑑/𝑑𝑡=𝐷 except at discontinuities.

The higher-order generalized invexity functions are defined as follows.

Definition 2.1. The scalar functional 𝐹(𝑥,𝑢)=𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡 is said to be higher-order 𝜌(𝜂,𝜉,𝜃)-invex in 𝑥, ̇𝑥, and 𝑢 if there exist 𝜂,𝜃𝐼×𝑛×𝑛×𝑛×𝑛×𝑚×𝑚𝑛,𝜉𝐼×𝑛×𝑛×𝑛×𝑛×𝑚×𝑚𝑚, 𝐼×𝑛×𝑛×𝑚×𝑛,1𝐼×𝑛×𝑛×𝑚×𝑚 and 𝜌, such that 𝐹(𝑥,𝑢)𝐹(𝑦,𝑣)𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝜌𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))2𝑑𝑡.(6)

Definition 2.2. The scalar functional 𝐹(𝑥,𝑢)=𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡 is said to be higher-order 𝜌(𝜂,𝜉,𝜃)-pseudoinvex in 𝑥, ̇𝑥, and 𝑢 if there exist 𝜂,𝜃𝐼×𝑛×𝑛×𝑛×𝑛×𝑚×𝑚𝑛,𝜉𝐼×𝑛×𝑛×𝑛×𝑛×𝑚×𝑚𝑚, 𝐼×𝑛×𝑛×𝑚×𝑛,1𝐼×𝑛×𝑛×𝑚×𝑚 and 𝜌, such that 𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝜌𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))2𝑑𝑡0𝐹(𝑥,𝑢)𝐹(𝑦,𝑣)𝑏𝑎(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡0.(7)

Definition 2.3. The scalar functional 𝐹(𝑥,𝑢)=𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡 is said to be higher-order 𝜌(𝜂,𝜉,𝜃)-quasi-invex in 𝑥, ̇𝑥 and 𝑢, if there exist 𝜂,𝜃𝐼×𝑛×𝑛×𝑛×𝑛×𝑚×𝑚𝑛, 𝜉𝐼×𝑛×𝑛×𝑛×𝑛×𝑚×𝑚𝑚, 𝐼×𝑛×𝑛×𝑚×𝑛, 1𝐼×𝑛×𝑛×𝑚×𝑚 and 𝜌, such that 𝐹(𝑥,𝑢)𝐹(𝑦,𝑣)𝑏𝑎(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡0𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝜌𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))2𝑑𝑡0.(8)

Remark 2.4. If (𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=𝑝𝑇𝑓𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑓𝑑𝑡̇𝑥,(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑞𝑇𝑓𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡)),(9) with 𝜂=0 at 𝑡=𝑎 and 𝑡=𝑏, then the above definitions becomes the definitions of invexity defined by Padhan and Nahak [10].

3. Mangasarian Type Higher-Order Duality

In this section, we propose the following Mangasarian type higher-order dual (MHCD) to (CP): (MHCD)max𝑏𝑎𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛼(𝑡)𝑇𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛽(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡,(10)subjectto𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑝𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑝𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=0,(11)𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑞𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑞𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=0,(12)𝑦(𝑎)=𝛾1,𝑦(𝑏)=𝛾2;𝑣(𝑎)=𝛿1,𝑣(𝑏)=𝛿2,(13)𝛼(𝑡)𝑟+,𝛽(𝑡)𝑠,𝛼1(𝑡)𝑟1+,𝛽1(𝑡)𝑠1,𝑝𝑛,𝑞𝑚.(14)

Remark 3.1. If 𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=𝑝𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑓𝑑𝑡̇𝑥+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑝𝑓𝑥𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑓𝑑𝑡𝑥̇𝑥(+𝑑𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑑𝑡2𝑓̇𝑥̇𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑝,1𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑞𝑢+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑞𝑓𝑢𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+2𝑓𝑥𝑢𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑞,𝑗𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=𝑝𝑗𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑔𝑑𝑡𝑗̇𝑥+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑝𝑔𝑗𝑥𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑔𝑑𝑡𝑗𝑥̇𝑥+𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑑𝑡2𝑔𝑗̇𝑥̇𝑥𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑝,𝑗=1,2,3,,𝑟,𝑗1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑞𝑔𝑗𝑢+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑞𝑔𝑗𝑢𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+2𝑔𝑗𝑥𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑞,𝑗=1,2,,𝑟1,𝑙𝑖𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=𝑝𝑖𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝐺𝑑𝑡𝑖̇𝑥+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑝𝐺𝑖𝑥𝑥𝑑(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝐺𝑑𝑡𝑖𝑥̇𝑥(+𝑑𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑑𝑡2𝐺𝑖̇𝑥̇𝑥𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑝,𝑖=1,2,3,,𝑠,𝑖1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑞𝐺𝑖𝑢+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))2𝑞𝐺𝑖𝑢𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+2𝐺𝑖𝑥𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑞,𝑖=1,2,3,,𝑠1,(15) then (MHCD) is similar to the second-order duality given by Padhan and Nahak [10].

Theorem 3.2 (weak duality). Let (𝑥(𝑡),𝑢(𝑡)) and (𝑦(𝑡),𝑣(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝,𝑞) be the feasible solutions of (CP) and (MHCD), respectively. Let 𝑏𝑎𝑓(𝑡,,,)𝑑𝑡, 𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡 and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡 be higher-order 𝜌0(𝜂,𝜉,𝜃)-invex, higher-order 𝜌1(𝜂,𝜉,𝜃)-invex, and higher-order 𝜌2(𝜂,𝜉,𝜃)-invex functions in 𝑥, ̇𝑥, and 𝑢 on 𝐼 with respect to the same functions 𝜂,𝜉,𝜃, with 𝜌0+𝜌1+𝜌20, then the following inequality holds between the primal (CP) and the dual (MHCD): 𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛼(𝑡)𝑇𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛽(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡.(16)

Proof. Since (𝑥(𝑡),𝑢(𝑡)) and (𝑦(𝑡),𝑣(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝,𝑞) be the feasible solutions of (CP) and (MHCD), respectively, we have 𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛼(𝑡)𝑇𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛽(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎+𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑑𝑡𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝛼(𝑡)𝑇+𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑑𝑡𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑑𝑡𝑏𝑎(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡0.(17) (by the higher-order 𝜌(𝜂,𝜉,𝜃)-invexity of 𝑏𝑎𝑓(𝑡,,,)𝑑𝑡,𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡) and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡,𝜌0+𝜌1+𝜌20 and equations (11)–(14)). This completes the proof.

We construct the following example which verifies Theorem 3.2 above, in which the objective and the constraints functions are higher-order 𝜌(𝜂,𝜉,𝜃)-invex.

Example 3.3. Let us consider the following control problem: min𝑏𝑎𝑥2(𝑡)𝑥(𝑡)𝑢(𝑡)𝑑𝑡,(18)subjectto𝑢(𝑡)𝑥2(𝑡)0,𝑥(𝑡)𝑢2(𝑡)=0,𝑥(𝑎)=𝛾1,𝑥(𝑏)=𝛾2,𝑢(𝑎)=𝛿1,𝑢(𝑏)=𝛿2,(19) where 𝐼=[𝑎,𝑏],𝑓𝐼××,𝑔𝐼×× and 𝐺𝐼××, 𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=𝑥2(𝑡)𝑥(𝑡)𝑢(𝑡), 𝑔(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=𝑢(𝑡)𝑥2(𝑡) and 𝐺(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=𝑥(𝑡)𝑢2(𝑡). It is clear that the objective function 𝑓, the inequality constraint function 𝑔 and the equality constraint function 𝐺 are not 1(𝜂,𝜉,𝜃)-invex, (1/2)(𝜂,𝜉,𝜃)-invex and (1/2)(𝜂,𝜉,𝜃)-invex, respectively, as defined by Padhan and Nahak [10]. But for 𝑢(𝑡)𝑣(𝑡)>1 and 𝑦(𝑡)>𝑣(𝑡), they are higher-order 1(𝜂,𝜉,𝜃)-invex, higher-order (1/2)(𝜂,𝜉,𝜃)-invex and higher-order (1/2)(𝜂,𝜉,𝜃)-invex, respectively, with respect to the same 𝜂,𝜉 and 𝜃. The functions 𝜂,𝜉,𝜃𝐼××××××, 𝐼××××, 𝑘𝐼××××, 𝑙𝐼××××,1𝐼××××, 𝑘1𝐼×××× and 𝑙1𝐼×××× are defined as follows: 𝑥𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))=2,(𝑡)+𝑥(𝑡)+𝑢(𝑡)+𝑦(𝑡)𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))=2𝑥2,=(𝑡)𝑦(𝑡)+2𝑥(𝑡)𝑦(𝑡)+2𝑦(𝑡)𝑢(𝑡)𝑢(𝑡)+𝑣(𝑡)𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑥2(𝑡)+𝑦2(𝑡),(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=(2𝑦(𝑡)1)𝑝𝑦2(𝑡)𝑣2(𝑡),𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=2𝑦(𝑡)𝑝𝑣2(𝑡),𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=𝑝𝑦2(𝑡)𝑣2(𝑡),1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑞𝑦2𝑘(𝑡),1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑞𝑣2𝑙(𝑡),1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑦2(𝑡)𝑣2(𝑡).(20) The Mangasarian higher-order dual of the control problem is max𝑏𝑎4𝑦2(𝑡)6𝑣2(𝑡)𝑑𝑡,(21)subjectto𝑦(𝑎)=𝛾1,𝑦(𝑏)=𝛾2;𝑣(𝑎)=𝛿1,𝑣(𝑏)=𝛿2,𝛼(𝑡)=𝛽(𝑡)=𝛼1(𝑡)=𝛽1(𝑡)=1.(22) The above problem satisfies weak duality Theorem 3.2 for 𝑥2(𝑡)+4𝑦2(𝑡)+6𝑣2(𝑡)𝑥(𝑡)+𝑢(𝑡).

Necessary conditions for the existence of an extremal solution for a variational problem subject to the both equality and inequality constraints were given by Valentine [11]. Using Valentine's results, Berkovitz [12] obtained the corresponding necessary conditions for the control problem (CP). These may be stated in the following way. If (𝑦(𝑡),𝑣(𝑡)) is an optimal solution for (CP), then𝜇0𝑓𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝑔𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑇𝛼(𝑡)+𝐺𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑇=𝑑𝛽(𝑡)𝜇𝑑𝑡0𝑓̇𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝑔̇𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑇𝛼(𝑡)+𝐺̇𝑥(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑇,𝜇𝛽(𝑡)(23)0𝑓𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝑔𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑇𝛼(𝑡)+𝑢(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝑇𝛽(𝑡)=0,(24)𝛼(𝑡)𝑇𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))=0,(25)𝛽(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))=0,(26)𝛼(𝑡)𝑟+(27) hold throughout 𝑎𝑡𝑏 (except for the values of 𝑡 corresponding to points of discontinuity of 𝑢(𝑡), (23) holds for right and left hand limits). Here, 𝜇0 is nonnegative constant, 𝛽(𝑡) is continuous in 𝑎𝑡𝑏, and 𝜇0,𝛼(𝑡), and 𝛽(𝑡) cannot vanish simultaneously for any 𝑎𝑡𝑏. It will be assumed that the minimizing arc determined by 𝑦(𝑡),𝑣(𝑡) is normal, that is, that 𝜇0 can be taken equal to 1.

Theorem 3.4 (strong duality). Let (𝑥(𝑡),𝑢(𝑡)) be a local or global optimal solution of (𝐶𝑃) at which the constraint qualification (23)–(27) are satisfied, and for the piecewise smooth functions 𝛼𝐼𝑟, 𝛽𝐼𝑠, let (i)(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=1(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=0,(ii)𝑘(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑘1(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=0,(iii)𝑙(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑙1(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=0,(iv)𝑝(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑓𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))(𝑑/𝑑𝑡)𝑓̇𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)),(v)𝑝𝛼(𝑡)𝑇𝑘(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0) = 𝑔𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛼(𝑡)(𝑑/𝑑𝑡)𝑔̇𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛼(𝑡),(vi)𝑝𝛽(𝑡)𝑇𝑙(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)= 𝐺𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛽(𝑡)(𝑑/𝑑𝑡)𝐺̇𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛽(𝑡), (vii)𝑞1(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑓𝑢(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)),(viii)𝑞𝛼(𝑡)𝑇𝑘1(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑔𝑢(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛼(𝑡),(ix)𝑞𝛽(𝑡)𝑇𝑙1(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)= 𝐺𝑢(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛽(𝑡).Then, (𝑥(𝑡),𝑢(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝=0,𝑞=0) is feasible for (MHCD). Moreover, if the weak duality Theorem 3.2 holds between the control primal (𝐶𝑃) and the Mangasarian higher-order dual (MHCD). Then, (𝑥(𝑡),𝑢(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝=0,𝑞=0) is an optimal solution of (MHCD), and the optimal values of (CP) and (MHCD) are equal.

Proof. Since (𝑥(𝑡),𝑢(𝑡)) is an optimal solution of (CP), from (23)–(27), we can easily conclude that (𝑥(𝑡),𝑢(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝=0,𝑞=0) satisfies the constraints of (MHCD) and objective values of (CP) and (MHCD) are equal. Hence, the result follows.

Theorem 3.5 (converse duality). Let (𝑥(𝑡),𝑢(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝,𝑞) be an optimal solution of (MWHCD). Suppose that 𝑏𝑎𝑓(𝑡,,,)𝑑𝑡, 𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡 and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡 are higher-order 𝜌0(𝜂,𝜉,𝜃)-invex, higher-order 𝜌1(𝜂,𝜉,𝜃)-invex, and higher-order 𝜌2(𝜂,𝜉,𝜃)-invex functions in 𝑥, ̇𝑥, and 𝑢 on 𝐼 with respect to the same functions 𝜂,𝜉,𝜃, with 𝜌0+𝜌1+𝜌20. Moreover, if 𝛼(𝑡)𝑇𝑔𝑡,𝑥(𝑡),𝑥(𝑡),+𝑢(𝑡)𝛽(𝑡)𝑇𝐺𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)×𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞+𝛼𝑘𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+𝛼1𝑘1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞+𝛽𝑙𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+𝛽1𝑙1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞0,(28) then (𝑥(𝑡),𝑢(𝑡)) is an optimal solution of (CP).

Proof. Suppose that (𝑥(𝑡),𝑢(𝑡)) is not an optimal solution of (CP). Then, there exists a feasible solution (𝑥(𝑡),𝑢(𝑡)) of the primal (CP) such that 𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡<𝑏𝑎𝑓𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)𝑑𝑡.(29) Since𝑏𝑎𝑓(𝑡,,,)𝑑𝑡, 𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡 and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡 are higher-order 𝜌0(𝜂,𝜉,𝜃)-invex, higher-order 𝜌1(𝜂,𝜉,𝜃)-invex, and higher-order 𝜌2(𝜂,𝜉,𝜃)-invex functions with respect to same 𝜂, 𝜉 and 𝜃, we have 𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝑓𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)𝑑𝑡𝑏𝑎𝛼(𝑡)𝑇𝑔𝑡,̇𝑥(𝑡),𝑥(𝑡),+𝑢(𝑡)𝛽(𝑡)𝑇𝐺𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)+𝑡,𝑥̇(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞+𝛼𝑘𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+𝛼1𝑘1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞+𝛽𝑙𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+𝛽1𝑙1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞𝑑𝑡(byequations(2),(3),(11),(12),and(14))0(byequations(28)).(30) This completes the proof.

Mond-Weir type higher-order duality is established to weaken the higher-order invexity requirements, that is, higher-order pseudoinvexiy and higher-order quasi-invexity.

4. Mond-Weir Type Higher-Order Duality

In this section, we propose the following Mond-Weir type higher-order dual (MWHCD) to (CP): (MWHCD)max𝑏𝑎𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡,(31)subjectto𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑝𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑝𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=0,(32)𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑞𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑞𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=0,(33)𝛼(𝑡)𝑇𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞𝛼1(𝑡)𝑇𝑘1𝛽(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)0,(34)1(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)0,(35)𝑦(𝑎)=𝛾1,𝑦(𝑏)=𝛾2,𝑣(𝑎)=𝛿1,𝑣(𝑏)=𝛿2,(36)𝛼(𝑡)𝑟+,𝛽(𝑡)𝑠,𝛼1(𝑡)𝑟1+,𝛽1(𝑡)𝑠1,𝑝𝑛,𝑞𝑚.(37)

Theorem 4.1 (weak duality). Let (𝑥(𝑡),𝑢(𝑡)) and (𝑦(𝑡),𝑣(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝,𝑞) be the feasible solutions of (CP) and (MWHCD), respectively. Let 𝑏𝑎𝑓(𝑡,,,)𝑑𝑡, 𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡 and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡 be higher-order 𝜌0(𝜂,𝜉,𝜃)-pseudoinvex, higher-order 𝜌1(𝜂,𝜉,𝜃)-quasi-invex, and higher-order 𝜌2(𝜂,𝜉,𝜃)-quasi-invex functions in 𝑥, ̇𝑥, and 𝑢 on 𝐼 with respect to the same functions 𝜂,𝜉, and 𝜃, with 𝜌0+𝜌1+𝜌20, then the following inequality holds between the primal (CP) and the dual (MHCD), 𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡.(38)

Proof. Since (𝑥(𝑡),𝑢(𝑡)) and (𝑦(𝑡),𝑣(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝,𝑞) are the feasible solutions of (CP) and (MWHCD), respectively, from (46), (3), (34), (35) and (37), we have 𝛼(𝑡)𝑇𝑔(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝛼(𝑡)𝑇𝑔(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑝𝑇𝑝𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑞𝑇𝑞𝛼1(𝑡)𝑇𝑘1𝛽(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)0,(39)1(𝑡)𝑇𝐺(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝛽1(𝑡)𝑇𝐺(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑝𝑇𝑝𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑞𝑇𝑞𝛽1(𝑡)𝑇𝑙1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)0.(40) Since 𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡 and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡 are higher-order 𝜌1(𝜂,𝜉,𝜃)-quasi-invex and higher-order 𝜌2(𝜂,𝜉,𝜃)-quasi-invex functions, (39) and (40) gives 𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑝𝛼(𝑡)𝑇𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝑝𝛽(𝑡)𝑇𝑙(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇+𝑞𝛼1(𝑡)𝑇𝑘1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝑞𝛽1(𝑡)𝑇𝑙1+𝜌(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)1+𝜌2(𝜃𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))2𝑑𝑡0𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)+𝜌𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))2𝑑𝑡0byequations(32),(33)and𝜌0+𝜌1+𝜌20𝑏𝑎𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))𝑑𝑡𝑏𝑎𝑓(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡))+(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)+1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑝𝑇𝑝(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)𝑞𝑇𝑞1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)𝑑𝑡0.(41) by the higher-order  𝜌0(𝜂,𝜉,𝜃)-pseudoinvexity of 𝑏𝑎𝑓(𝑡,,,)𝑑𝑡. This completes the proof.

We construct the following example which verifies Theorem 4.1 above, in which the objective function is higher-order 𝜌(𝜂,𝜉,𝜃)-pseudoinvex and the constraints functions are higher-order 𝜌(𝜂,𝜉,𝜃)-quasi-invex.

Example 4.2. Let us consider the following control problem: min𝑏𝑎𝑥2(𝑡)+𝑢2(𝑡)+𝑥(𝑡)+𝑢(𝑡)𝑑𝑡,(42)subjectto𝑥2𝑢(𝑡)𝑥(𝑡)𝑢(𝑡)0,2(𝑡)𝑥(𝑡)=0,𝑥(𝑎)=𝛾1,𝑥(𝑏)=𝛾2,𝑢(𝑎)=𝛿1,𝑢(𝑏)=𝛿2,(43) where 𝐼=[𝑎,𝑏],𝑓𝐼××,𝑔𝐼××,𝐺𝐼××, 𝑓(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=𝑥2(𝑡)+𝑢2(𝑡)+𝑥(𝑡)+𝑢(𝑡), 𝑔(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=𝑥2(𝑡)𝑥(𝑡)𝑢(𝑡), and 𝐺(𝑡,𝑥(𝑡),̇𝑥(𝑡),𝑢(𝑡))=𝑢2(𝑡)𝑥(𝑡). It is clear that the objective function 𝑓, the inequality constraint function 𝑔 and the equality constraint function 𝐺 are not higher-order -(1/2)-(𝜂,𝜉,𝜃)-invex, higher-order 1(𝜂,𝜉,𝜃)invex and higher-order -(1/2)-(𝜂,𝜉,𝜃)-invex, respectively. But they are higher-order -(1/2)-(𝜂,𝜉,𝜃)-pseudoinvex, higher-order 1-(𝜂,𝜉,𝜃)-quasi-invex and higher-order -(1/2)-(𝜂,𝜉,𝜃)-quasi-invex, respectively, with respect to the same 𝜂,𝜉 and 𝜃. The functions 𝜂,𝜉,𝜃𝐼××××××,𝐼××××, 𝑘𝐼××××, 𝑙𝐼××××,1𝐼××××,𝑘1𝐼××××, and 𝑙1𝐼×××× are defined as follows: 𝑥𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))=2,𝑢(𝑡)+𝑢(𝑡)+1𝜉(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))=2,=(𝑡)+𝑥(𝑡)+1𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),̇𝑥(𝑡),̇𝑦(𝑡),𝑢(𝑡),𝑣(𝑡))𝑥2(𝑡)𝑦2(𝑡)+𝑢(𝑡)𝑦2(𝑡)+𝑢2(𝑡)𝑣2(𝑡)+𝑥(𝑡)𝑣2𝑦(𝑡),(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=2(𝑡)+𝑣2𝑦(𝑡)2(𝑡)+𝑣2𝑝,(𝑡)+1𝑘(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑝)=𝑦(𝑡)𝑣(𝑡)+𝑣2(𝑡)+𝑦2(𝑡)𝑝,1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=2𝑣2𝑦(𝑡)+𝑦(𝑡)+2(𝑡)+𝑣2𝑞,(𝑡)+11(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=2𝑣2(𝑦𝑡)+𝑦(𝑡)+2(𝑡)+𝑣2(𝑞,𝑘𝑡)+11(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=𝑦2(𝑡)+𝑣2𝑙(𝑡)𝑞,1(𝑡,𝑦(𝑡),̇𝑦(𝑡),𝑣(𝑡),𝑞)=4𝑦2(𝑡)+5𝑣2𝑦(𝑡)+2(𝑡)+1𝑞.(44) The Mond-Weir higher-order dual of the control problem is max𝑏𝑎𝑦2(𝑡)+𝑣2(𝑡)𝑑𝑡,(45)subjectto2𝑦2(𝑡)+𝑣2(𝑡)0,(46)6𝑦2(𝑡)+9𝑣2(𝑡)0,(47)𝑦(𝑎)=𝛾1,𝑦(𝑏)=𝛾2,𝑣(𝑎)=𝛿1,𝑣(𝑏)=𝛿2,(48)𝛼(𝑡)=𝛽(𝑡)=𝛼1(𝑡)=𝛽1(𝑡)=1.(49) The above problem satisfies weak duality Theorem 4.1.

Theorem 4.3 (strong duality). Let (𝑥(𝑡),𝑢(𝑡)) be a local or global optimal solution of (CP), and for the piecewise smooth functions 𝛼𝐼𝑟, 𝛽𝐼𝑠, let (i)(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=1(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=0,(ii)𝑘(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑘1(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=0,(iii)𝑙(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑙1(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=0,(iv)𝑝(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑓𝑥(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))(𝑑/𝑑𝑡)𝑓̇𝑥(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)),(v)𝑝𝛼(𝑡)𝑇𝑘(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0) = 𝑔𝑥(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛼(𝑡)(𝑑/𝑑𝑡)𝑔̇𝑥(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛼(𝑡),(vi)𝑝𝛽(𝑡)𝑇𝑙(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0) = 𝐺𝑥(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛽(𝑡)(𝑑/𝑑𝑡)𝐺̇𝑥(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛽(𝑡),(vii)𝑞1(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑓𝑢(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)),(viii)𝑞𝛼(𝑡)𝑇𝑘1(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0) = 𝑔𝑢(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛼(𝑡),(ix)𝑞𝛽(𝑡)𝑇𝑙1(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0) = 𝐺𝑢(𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))𝑇𝛽(𝑡).Then, (𝑥(𝑡),𝑢(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝=0,𝑞=0) is feasible for (MWHCD). Moreover, if the Weak Duality Theorem 4.1 holds between the control primal (CP) and the Mond-Weir higher-order dual (MWHCD). Then, (𝑥(𝑡),𝑢(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝=0,𝑞=0) is an optimal solution of (MWHCD), and the optimal values of (CP) and (MHCD) are equal.

Proof. The proof is similar to that of Theorem 3.4.

Theorem 4.4 (converse duality). Let (𝑦(𝑡),𝑣(𝑡),𝛼(𝑡),𝛽(𝑡),𝑝,𝑞) be an optimal solution of (MWHCD). Suppose that 𝑏𝑎𝑓(𝑡,,,)𝑑𝑡, 𝑏𝑎𝛼(𝑡)𝑇𝑔(𝑡,,,)𝑑𝑡 and 𝑏𝑎𝛽(𝑡)𝑇𝐺(𝑡,,,)𝑑𝑡 are higher-order 𝜌0(𝜂,𝜉,𝜃)-pseudoinvex, higher-order 𝜌1(𝜂,𝜉,𝜃)-quasi-invex, and higher-order 𝜌2(𝜂,𝜉,𝜃)-quasi-invex functions in 𝑥, 𝑥 and 𝑢 on 𝐼 with respect to the same functions 𝜂,𝜉,𝜃, with 𝜌0+𝜌1+𝜌20. Moreover, if 𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝+1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞𝑝𝑇𝑝𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑝𝑞𝑇𝑞1𝑡,̇𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),𝑞0,(50) then (𝑥(𝑡),𝑢(𝑡)) is an optimal solution of (CP).

Proof. The proof is similar to that of Theorem 3.5.

5. Concluding Remarks

In this paper, we have studied both Mangasarian type and Mond-Weir type higher-order duality of the control problems. By taking different examples, it is verified that our higher-order generalized invexity is more general than the existing definitions of invexity in the literature. Example 3.3 shows that the objective and constraint functions of the (CP) are not invex as defined by Padhan and Nahak [10], but they are higher-order 𝜌(𝜂,𝜉,𝜃)-invex and also satisfy weak duality relation with Mangasarian type higher-order duality. In Example 4.2, the objective and constraint functions are not higher-order 𝜌(𝜂,𝜉,𝜃)-invex but the objective function is higher-order 𝜌(𝜂,𝜉,𝜃)-pseudoinvex and the constraint functions are higher-order 𝜌(𝜂,𝜉,𝜃)-quasi-invex and also satisfy weak duality relation with Mond-Weir higher-order duality.

In this paper, the objective function and the constraint functions are twice continuously differentiable. Relaxing this assumptions to include nonsmooth higher-order generalized invex functions for control problems is immediately a topic of further research.

References

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