Higher-Order Generalized Invexity in Control Problems

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 problems(uCbPje)cmtitno𝑥(𝑔𝐺𝑎^𝑏𝑎()(𝑡𝑡𝑓,=,𝑥𝑥(𝛾𝑡((,𝑡₁𝑡)𝑥),,(𝑥𝑡𝑥)̇(̇(,𝑡𝑡)𝑥),̇,(𝑢𝑡𝑢)((𝑡,𝑡)𝑢)))(≤𝑡=))00𝑑,,𝑡,(((123)))𝑢(𝑎)=𝛿₁,𝑥(𝑏)=𝛾₂;,𝑢(𝑏)=𝛿₂,(4) where 𝑓, 𝑔, and 𝐺 are twice continuously differentiable functions from 𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚 into ℝ, ℝ^𝑟, and ℝ^𝑠, respectively, 𝐼=[𝑎,𝑏]⊆ℝ.

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 (𝐹−𝜌) 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,ℝ^𝑛) denote the space of piecewise smooth functions 𝑥 with norm ‖𝑥‖=‖𝑥‖_∞+‖𝐷𝑥‖_∞, where the differentiation operator 𝐷 is given by𝑢=𝐷𝑥⟺𝑥(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),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡 is said to be higher-order 𝜌−(𝜂,𝜉,𝜃)-invex in 𝑥, 𝑥̇, and 𝑢 if there exist 𝜂,𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ^𝑛,𝜉∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ^𝑚, ℎ∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑛→ℝ,ℎ₁∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ and 𝜌∈ℝ, such that 𝐹(≥𝑥,𝑢)−𝐹(𝑦,𝑣)_𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇∇+_𝑝ℎ𝜉((𝑡𝑡,,𝑥𝑦((𝑡𝑡)),,𝑦𝑦(̇(𝑡𝑡)),,𝑥𝑣̇((𝑡𝑡)),,𝑦𝑝̇)(𝑡),𝑢(𝑡),𝑣(𝑡))_𝑇∇++_𝑞ℎℎℎ₁₁(𝑡(,𝑡,𝑦𝑦(𝑡()𝑡),,𝑦𝑦̇(̇𝑡()𝑡),,𝑣𝑣(𝑡()𝑡),,𝑝𝑞))−𝑝_𝑇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝜌‖𝜃∇(_𝑞𝑡ℎ,₁𝑥((𝑡𝑡,)𝑦,(𝑦𝑡)(𝑡,)𝑦,̇(𝑥𝑡̇)(𝑡,)𝑣,(𝑦𝑡̇)(𝑡,)𝑞,)𝑢(𝑡),𝑣(𝑡))‖₂𝑑𝑡.(6)

Definition 2.2. The scalar functional 𝐹(𝑥,𝑢)=∫^𝑏𝑎𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡 is said to be higher-order 𝜌−(𝜂,𝜉,𝜃)-pseudoinvex in 𝑥, 𝑥,̇ and 𝑢 if there exist 𝜂,𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ^𝑛,𝜉∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ^𝑚, ℎ∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑛→ℝ,ℎ₁∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ and 𝜌∈ℝ, such that ^𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇∇+_𝑝ℎ𝜉((𝑡𝑡,,𝑥𝑦((𝑡𝑡)),,𝑦𝑦(̇(𝑡𝑡)),,𝑥𝑣̇((𝑡𝑡)),,𝑦𝑝̇)(𝑡),𝑢(𝑡),𝑣(𝑡))_𝑇∇+_𝑞ℎ𝜌₁‖(𝜃𝑡,(𝑦𝑡,(𝑥𝑡)(𝑡,)𝑦,̇(𝑦𝑡)(𝑡,)𝑣,(𝑥𝑡)̇(,𝑡)𝑞,)𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))‖₂⟹−𝐹(𝑥,𝑢)−𝐹(𝑦,𝑣)_𝑏𝑎𝑑𝑡≥0−ℎ𝑝_𝑇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+ℎ₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)∇_𝑞ℎ₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡≥0.(7)

Definition 2.3. The scalar functional 𝐹(𝑥,𝑢)=∫^𝑏𝑎𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡 is said to be higher-order 𝜌−(𝜂,𝜉,𝜃)-quasi-invex in 𝑥, 𝑥̇ and 𝑢, if there exist 𝜂,𝜃∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ^𝑛, 𝜉∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ^𝑚, ℎ∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑛→ℝ, ℎ₁∶𝐼×ℝ^𝑛×ℝ^𝑛×ℝ^𝑚×ℝ^𝑚→ℝ and 𝜌∈ℝ, such that 𝐹(𝑥,−𝑢)−𝐹(𝑦,𝑣)_𝑏𝑎ℎ+(ℎ𝑡,₁𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)−𝑝_𝑇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)⟹∇_𝑞ℎ₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡≤0_𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇∇+_𝑝ℎ𝜉((𝑡𝑡,,𝑥𝑦((𝑡𝑡)),,𝑦𝑦(̇(𝑡𝑡)),,𝑥𝑣̇((𝑡𝑡)),,𝑦𝑝̇)(𝑡),𝑢(𝑡),𝑣(𝑡))_𝑇∇+_𝑞𝜌ℎ‖₁𝜃(𝑡(,𝑡,𝑦𝑥(𝑡()𝑡),𝑦,𝑦̇(𝑡()𝑡),𝑣,𝑥(̇𝑡()𝑡,)𝑞,)𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))‖₂≤0.𝑑𝑡(8)

Remark 2.4. If ℎ(𝑡,𝑦=(𝑡𝑝)_𝑇,𝑦̇(𝑡),𝑣(𝑡),𝑝)ℎ𝑓^𝑥(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))−𝑑𝑑𝑡𝑓^̇𝑥(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))₁,(𝑡,=𝑦𝑞(𝑡_𝑇),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑓_𝑢(𝑡,𝑦(𝑡),𝑦̇(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)max^𝑏𝑎𝑓+(𝛼𝑡,(𝑦𝑡)(_𝑇𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝛽(𝑡)_𝑇𝑔(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))++ℎ𝛼((𝑡𝑡),_𝑇𝑦𝐺(𝑡()𝑡,,𝑦𝑦̇((𝑡𝑡)),,𝑣𝑦̇((𝑡𝑡)),,𝑝𝑣)(𝑡))+𝛽(𝑡)_𝑇𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+ℎ₁𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝛼₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝛽₁(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)subjectto∇_𝑝(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡,(10)+ℎ(∇𝑡,_𝑝𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+∇_𝑝𝛼(𝑡)^𝑇𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)∇𝛽(𝑡)^𝑇𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)=0,(11)_𝑞ℎ₁+(𝑡∇,𝑦_𝑞(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+∇_𝑞𝛼₁(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑦(𝑎𝛽)₁=(𝑡)𝛾^𝑇₁𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)=0,(12)𝑣(𝑎)=𝛿₁,𝑦(𝑏)=𝛾₂;𝛼(𝑡)∈ℝ_𝑟+,𝑣(𝑏)=𝛿₂,(13)𝛽₁,𝛽(𝑡)∈ℝ^𝑠,𝛼₁(𝑡)∈ℝ^𝑟1₊,(𝑡)∈ℝ^𝑠1,𝑝∈ℝ^𝑛,𝑞∈ℝ^𝑚.(14)

Remark 3.1. If ℎ(=𝑡,𝑦𝑝(𝑡)𝑓,_𝑥𝑦̇(𝑡),𝑣(𝑡),𝑝)(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))−𝑑𝑑𝑡𝑓^̇𝑥+1(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))2𝑝𝑓^𝑥𝑥−2𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑑𝑡𝑓^𝑥̇𝑥+𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))₂𝑑𝑡²ℎ𝑓_̇𝑥̇𝑥(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑝,₁=(𝑡,𝑞𝑦(𝑓𝑡)_𝑢,𝑦̇(𝑡),𝑣(𝑡),𝑞)+1(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))2𝑞𝑓^𝑢𝑢𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))+2𝑓_𝑥𝑢(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑞,_𝑗(=𝑡,𝑝𝑦(𝑡𝑔)_𝑗𝑥,𝑦̇(𝑡),𝑣(𝑡),𝑝)(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))−𝑑𝑑𝑡𝑔^𝑗̇𝑥+1(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))2𝑝𝑔^𝑗𝑥𝑥−2𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑑𝑡𝑔^{𝑗𝑥̇𝑥}+𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))₂𝑑𝑡²𝑘𝑔^{𝑗̇𝑥̇𝑥}𝑗(𝑡=,𝑦1(𝑡,)2,,𝑦3̇,(𝑡…),𝑣,𝑟(,_𝑗1𝑡))𝑝,=(𝑡,𝑞𝑦𝑔(_𝑗𝑢𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+1(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))2𝑞𝑔^𝑗𝑢𝑢+2𝑔(𝑡,_𝑗𝑥𝑢𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))(𝑡,𝑗𝑦=(𝑡)1,,𝑦2̇(,𝑡…),𝑣,𝑟(₁𝑡))𝑞,𝑙_𝑖,(=𝑡,𝑦𝑝(𝑡)𝐺,_𝑖𝑥𝑦̇(𝑡),𝑣(𝑡),𝑝)(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))−𝑑𝑑𝑡𝐺^𝑖̇𝑥+1(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))2𝑝𝐺^𝑖𝑥𝑥−2𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑑𝑡𝐺^{𝑖𝑥̇𝑥}+𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))₂𝑑𝑡²𝑙𝐺^{𝑖̇𝑥̇𝑥}(𝑡,𝑖𝑦=(𝑡1),,2𝑦,̇(3𝑡,)…,𝑣,(𝑡𝑠),_𝑖1)𝑝,(=𝑡,𝑦𝑞𝐺(𝑡_𝑖𝑢),𝑦̇(𝑡),𝑣(𝑡),𝑞)+1(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))2𝑞𝐺^𝑖𝑢𝑢+2𝐺(𝑡,_𝑖𝑥𝑢𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))(𝑡,𝑦(𝑡),𝑦̇(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+𝜌₂≥0, then the following inequality holds between the primal (CP) and the dual (MHCD): ^𝑏𝑎≥𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎+𝑓𝛽(𝑡(,𝑡)𝑦_𝑇(𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝛼(𝑡)^𝑇𝑔(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))++ℎℎ₁(𝑡,𝑦𝐺(𝑡()𝑡,,𝑦𝑦̇((𝑡𝑡)),,𝑣𝑦̇((𝑡𝑡)),,𝑝𝑣)(𝑡))+𝛼(𝑡()𝑡_𝑇,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝛼₁𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝛽(𝑡()𝑡_𝑇)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝛽₁𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(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),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎𝑓+(𝛽𝑡(,𝑡𝑦)_𝑇(𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝛼(𝑡)^𝑇𝑔(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))++ℎℎ₁(𝑡,𝑦𝐺(𝑡()𝑡,,𝑦𝑦̇((𝑡𝑡)),,𝑣𝑦̇((𝑡𝑡)),,𝑝𝑣)(𝑡))+𝛼(𝑡()𝑡_𝑇,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝛼₁𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝛽(𝑡()𝑡_𝑇)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝛽₁𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)≥(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡_𝑏𝑎−𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎+𝑓(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑑𝑡_𝑏𝑎−𝛼(𝑡)^𝑇𝑔(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎+𝛼(𝑡)^𝑇𝑔(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑑𝑡_𝑏𝑎−𝛽(𝑡)^𝑇𝐺(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎−𝛽(𝑡)^𝑇𝐺(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))𝑑𝑡_𝑏𝑎ℎ+(ℎ𝑡,₁𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝛼(𝑡()𝑡_𝑇,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝛼₁𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝛽(𝑡()𝑡_𝑇)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝛽₁𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)≥0.(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡(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+𝜌₂≥0 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: subjecmtitno𝑢^𝑏𝑎(𝑡)𝑥−²𝑥(𝑡₂)−𝑥(𝑡)−𝑢(𝑡)𝑑𝑡,(18)𝑥(𝑡)−𝑢₂(𝑡)≤0,𝑥(𝑎)=𝛾₁(𝑡)=0,𝑢(𝑎)=𝛿₁,𝑥(𝑏)=𝛾₂,,𝑢(𝑏)=𝛿₂,(19) where 𝐼=[𝑎,𝑏],𝑓∶𝐼×ℝ×ℝ→ℝ,𝑔∶𝐼×ℝ×ℝ→ℝ and 𝐺∶𝐼×ℝ×ℝ→ℝ, 𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))=𝑥²(𝑡)−𝑥(𝑡)−𝑢(𝑡), 𝑔(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))=𝑢(𝑡)−𝑥²(𝑡) and 𝐺(𝑡,𝑥(𝑡),𝑥̇(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,𝜃∶𝐼×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ→ℝ, ℎ∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ, 𝑘∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ, 𝑙∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ,ℎ₁∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ, 𝑘₁∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ and 𝑙₁∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ are defined as follows: 𝜂(𝑡,𝑥=(𝑡−),𝑦𝑥(₂𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))𝜉(𝑡,𝑥=(𝑡−),𝑦2(𝑥𝑡₂()𝑡,)𝑥+̇(𝑡𝑥),(𝑡𝑦)̇(+𝑡)𝑢,𝑢(𝑡()𝑡)+,𝑣𝑦((𝑡𝑡))),𝜃(𝑡,𝑥=(𝑡√),𝑦(𝑡)(,𝑡𝑥)̇𝑦(𝑡()𝑡,)𝑦+̇(𝑡2)𝑥,𝑢(𝑡()𝑡𝑦),(𝑣𝑡)(+𝑡))2𝑦(𝑡)𝑢(𝑡)−𝑢(𝑡)+𝑣(𝑡),ℎ(𝑡,𝑦=(𝑡()2,𝑦𝑥𝑦²(̇𝑡(()𝑡𝑡))−,+𝑣1()𝑦𝑡𝑝²)(,−𝑡𝑝)),𝑦₂𝑘(𝑡,𝑦=(𝑡−)2,𝑦𝑦̇((𝑡𝑡)),𝑝𝑣−(𝑡𝑣)₂,𝑝)(𝑡)−𝑣²(𝑡),𝑙(𝑡,𝑦=(𝑡𝑝),−𝑦̇𝑦(𝑡₂),𝑣(𝑡),𝑝()𝑡),ℎ(𝑡)−𝑣²(𝑡),₁(𝑡,=𝑦−(𝑡)𝑞,−𝑦̇(𝑦𝑡₂),𝑣(𝑡),𝑞)𝑘(𝑡),₁(𝑡,=𝑦(𝑞𝑡)−,𝑦𝑣₂̇(𝑡),𝑣(𝑡),𝑞)𝑙(𝑡),₁(𝑡,=𝑦(−𝑡)𝑦,₂𝑦̇(𝑡),𝑣(𝑡),𝑞)(𝑡)−𝑣²(𝑡).(20) The Mangasarian higher-order dual of the control problem is subjemctatxo𝑦^𝑏𝑎(𝑎)−=4𝑦𝛾²₁(𝑡)−6𝑣²(𝑡)𝑑𝑡,(21)𝑣(𝑎)=𝛿₁,𝑦(𝑏)=𝛾₂;𝛼(𝑡)=𝛽(,𝑡)=𝛼𝑣₁(𝑏)=𝛿₂,(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), then𝜇₀𝑓_𝑥(𝑡+,𝑦𝐺(𝑡_𝑥),𝑦̇(𝑡),𝑣(𝑡))+𝑔_𝑥(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))^𝑇𝛼(𝑡)=𝑑(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))^𝑇𝛽(𝑡)𝑑𝑡𝜇⁰+𝑔𝑓_̇𝑥̇𝑥(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝐺_̇𝑥(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))^𝑇𝛼(𝑡)𝜇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))^𝑇𝛽(𝑡),(23)₀𝑓_𝑢+(𝑡ℎ,𝑦_𝑢(𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝑔_𝑢(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))^𝑇𝛼(𝑡)(𝑡,𝑦𝛼((𝑡𝑡))_𝑇,𝑦̇(𝑡),𝑣(𝑡))^𝑇𝛽(𝑡)=0,(24)𝛽(𝑡)_𝑇𝑔(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))=0,(25)𝛼𝐺(𝑡)(𝑡∈,𝑦ℝ(𝑡_𝑟+),𝑦̇(𝑡),𝑣(𝑡))=0,(26)(27) hold throughout 𝑎≤𝑡≤𝑏 (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 𝑎≤𝑡≤𝑏, and 𝜇₀,𝛼(𝑡), and 𝛽(𝑡) cannot vanish simultaneously for any 𝑎≤𝑡≤𝑏. 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 𝛼∶𝐼→ℝ^𝑟, 𝛽∶𝐼→ℝ^𝑠, let (i)ℎ(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=ℎ₁(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)∇_𝑝ℎ(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑓_𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))−(𝑑/𝑑𝑡)𝑓_̇𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)),(v)∇_𝑝𝛼(𝑡)^𝑇𝑘(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0) = 𝑔_𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))^𝑇𝛼(𝑡)−(𝑑/𝑑𝑡)𝑔_̇𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))^𝑇𝛼(𝑡),(vi)∇_𝑝𝛽(𝑡)^𝑇𝑙(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)= 𝐺_𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))^𝑇𝛽(𝑡)−(𝑑/𝑑𝑡)𝐺_̇𝑥(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))^𝑇𝛽(𝑡), (vii)∇_𝑞ℎ₁(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑓_𝑢(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)),(viii)∇_𝑞𝛼(𝑡)^𝑇𝑘₁(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)=𝑔_𝑢(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡))^𝑇𝛼(𝑡),(ix)∇_𝑞𝛽(𝑡)^𝑇𝑙₁(𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡),0)= 𝐺_𝑢(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, 𝑥̇‌, and 𝑢 on 𝐼 with respect to the same functions 𝜂,𝜉,𝜃, with 𝜌₀+𝜌₁+𝜌₂≥0. Moreover, if 𝛼(𝑡)^𝑇𝑔×𝑡ℎ,𝑥(𝑡𝑡,),𝑥(𝑡),𝑢(𝑡)+𝛽(𝑡)^𝑇𝐺𝑡,𝑥(𝑡),𝑥(𝑡),𝑢(𝑡)+ℎ₁𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝+𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞+𝛼𝑘𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝+𝛼₁𝑘₁𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞+𝛽𝑙𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝≥0,𝛽₁𝑙₁𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞(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),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡<^𝑏𝑎𝑓𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡)𝑑𝑡.(29) Since∫^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 with respect to same 𝜂, 𝜉 and 𝜃, we have ^𝑏𝑎≥𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡−^𝑏𝑎𝑓𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡)𝑑𝑡_𝑏𝑎+𝛼(𝑡)^𝑇𝑔𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡)+ℎ𝛽(𝑡)𝑡^𝑇,𝐺𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡)+ℎ₁𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝+𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞+𝛼𝑘𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝+𝛼₁𝑘₁𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞+𝛽𝑙𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝((bbyy≥ee0qquuaattiioo𝛽nn₁ss𝑙₁((22𝑡)8,𝑥,))((.3𝑡)),,𝑥(̇1(𝑡1)),,𝑢((1𝑡2)),𝑞,an𝑑d𝑡(14))(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)max^𝑏𝑎𝑓++(ℎℎ𝑡,₁(𝑦𝑡,(𝑦𝑡)(,𝑡𝑦)̇,(𝑦𝑡)̇(,𝑡𝑣),(𝑣𝑡))(𝑡),𝑝)−𝑝_𝑇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)subjectto∇_𝑝∇_𝑞ℎ₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡,(31)+ℎ(∇𝑡,_𝑝𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+∇_𝑝𝛼(𝑡)^𝑇𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)∇=0,𝛽(𝑡)^𝑇𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)(32)_𝑞ℎ₁+(𝑡∇,𝑦_𝑞(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+∇_𝑞𝛼₁(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝛼(𝑡=)_𝑇0,𝛽₁(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)(33)+𝑔𝛼(𝑡(,𝑡𝑦)_𝑇(𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝛼₁𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)−𝑝_𝑇(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝𝛼(𝑡)^𝑇𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)𝛽≥0,∇_𝑞𝛼₁(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)(34)₁(𝑡)+^𝑇𝐺𝛽((𝑡𝑡),_𝑇𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))+𝛽₁𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)−𝑝_𝑇(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝𝛽(𝑡)^𝑇𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)𝑦(𝑎≥)0=,𝛾₁∇_𝑞𝛽₁(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)(35)𝑣(𝑎)=𝛿₁,𝑦(𝑏)=𝛾₂,𝛼(𝑡)∈ℝ_𝑟+,𝑣(𝑏)=𝛿₂,(36)𝛽₁,𝛽(𝑡)∈ℝ^𝑠,𝛼₁(𝑡)∈ℝ^𝑟1₊,(𝑡)∈ℝ^𝑠1,𝑝∈ℝ^𝑛,𝑞∈ℝ^𝑚.(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, 𝑥̇, and 𝑢 on 𝐼 with respect to the same functions 𝜂,𝜉, and 𝜃, with 𝜌₀+𝜌₁+𝜌₂≥0, then the following inequality holds between the primal (CP) and the dual (MHCD), ^𝑏𝑎𝑓≥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎+𝑓ℎ(𝑡₁,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))+ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)−𝑝_𝑇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)∇_𝑞ℎ₁(𝑡,𝑦(𝑡),𝑦̇(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)^𝑇−𝑔𝛼(𝑡(,𝑡𝑥)_𝑇(𝑡),𝑥̇(𝑡),𝑢(𝑡))−𝛼(𝑡)^𝑇𝑔(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))−𝛼₁𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝑝_𝑇(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝑞_𝑇∇_𝑝𝛼(𝑡)^𝑇𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)𝛽∇_𝑞𝛼₁(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)≤0,(39)₁(𝑡)−^𝑇𝐺𝛽((𝑡𝑡),_𝑇𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))−𝛽₁(𝑡)^𝑇𝐺(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))−𝛽₁𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝑝_𝑇(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝑞_𝑇∇_𝑝𝛽(𝑡)^𝑇𝑙(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)∇_𝑞𝛽₁(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(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 ^𝑏𝑎𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇∇_𝑝𝛼+(𝑡∇)^𝑇_𝑝𝑘(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)+𝜉(𝑡,𝑥(𝑡𝛽)(,𝑡𝑦)^𝑇(𝑙𝑡)(𝑡,,𝑥𝑦̇((𝑡𝑡)),,𝑦𝑦̇(̇(𝑡𝑡)),,𝑢𝑣((𝑡𝑡)),,𝑣𝑝()𝑡))_𝑇+∇+_𝑞∇𝛼_1𝑞(𝑡)^𝑇𝑘₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)+𝜌𝛽₁(𝑡)^𝑇𝑙₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)₁⟹≤0_𝑏𝑎+𝜌₂‖𝜃(𝑡,𝑥(𝑡),𝑦(𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))‖²𝑑𝑡𝜂(𝑡,𝑥(𝑡),𝑦(𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))^𝑇∇+_𝑝ℎ𝜉((𝑡𝑡,,𝑥𝑦((𝑡𝑡)),,𝑦𝑦(̇(𝑡𝑡)),,𝑥𝑣̇((𝑡𝑡)),,𝑦𝑝̇)(𝑡),𝑢(𝑡),𝑣(𝑡))_𝑇∇+_𝑞𝜌ℎ‖₁𝜃(𝑡(,𝑡𝑦,𝑥(𝑡()𝑡,)𝑦,̇𝑦(𝑡()𝑡,)𝑣,𝑥(𝑡̇)(𝑡,)𝑞,)𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))‖₂byequations(32),(33)and𝜌₀𝑑𝑡≥0⟹+𝜌₁+𝜌₂≥0_𝑏𝑎−𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))𝑑𝑡_𝑏𝑎𝑓+(ℎ𝑡,₁𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡))+ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)−𝑝_𝑇(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)−𝑞_𝑇∇_𝑝ℎ(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑝)∇_𝑞ℎ₁(𝑡,𝑦(𝑡),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑑𝑡≥0.(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: subjecmtitno𝑥^𝑏𝑎₂𝑥²(𝑡)+𝑢²(𝑡)+𝑥(𝑡)+𝑢(𝑡)𝑑𝑡,(42)𝑢(𝑡)−𝑥(𝑡)𝑢(𝑡)≤0,₂𝑥((𝑎𝑡))=−𝛾𝑥₁(𝑡)=0,𝑢(𝑎)=𝛿₁,𝑥(𝑏)=𝛾₂,,𝑢(𝑏)=𝛿₂,(43) where 𝐼=[𝑎,𝑏],𝑓∶𝐼×ℝ×ℝ→ℝ,𝑔∶𝐼×ℝ×ℝ→ℝ,𝐺∶𝐼×ℝ×ℝ→ℝ, 𝑓(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))=𝑥²(𝑡)+𝑢²(𝑡)+𝑥(𝑡)+𝑢(𝑡), 𝑔(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))=𝑥²(𝑡)−𝑥(𝑡)𝑢(𝑡), and 𝐺(𝑡,𝑥(𝑡),𝑥̇(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,𝜃∶𝐼×ℝ×ℝ×ℝ×ℝ×ℝ×ℝ→ℝ,ℎ∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ, 𝑘∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ, 𝑙∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ,ℎ₁∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ,𝑘₁∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ, and 𝑙₁∶𝐼×ℝ×ℝ×ℝ×ℝ→ℝ are defined as follows: 𝜂(𝑡,𝑥=(𝑡−),𝑦𝑥(₂𝑡),𝑥̇(𝑡),𝑦̇(𝑡),𝑢(𝑡),𝑣(𝑡))𝜉(𝑡,𝑥=(𝑡−),𝑦𝑢₂(𝑡()𝑡,)𝑥+̇(𝑡𝑢)(,𝑡𝑦)̇(+𝑡)1,𝑢,(𝑡),𝑣(𝑡))𝜃(𝑡,𝑥=(𝑡√),𝑦(𝑡()𝑡),𝑥+̇(𝑥𝑡)(,𝑡𝑦)̇(+𝑡)1,𝑢,(𝑡),𝑣(𝑡))ℎ(𝑡,𝑦=(𝑡−),𝑥𝑦𝑦²̇₂((𝑡𝑡)),𝑦𝑣²((𝑡𝑡)),+𝑝)𝑢(𝑡)𝑦²(𝑡)+𝑢²(𝑡)𝑣²(𝑡)+𝑥(𝑡)𝑣²(𝑡),𝑘(𝑡,𝑦=(𝑡𝑦)(,𝑡𝑦)̇𝑣(𝑡()(𝑡𝑡,))𝑣++(𝑡𝑣𝑣)²₂,(𝑝𝑡))−𝑦²(𝑡)+𝑣²(𝑡)+1𝑝,ℎ(𝑡)+𝑦²(𝑡)𝑝,₁(𝑡,=𝑦−(𝑡),2𝑦𝑣̇(₂𝑡),𝑣(𝑡),𝑞)ℎ(𝑡)+𝑦(𝑡)+𝑦²(𝑡)+𝑣²(𝑡)+1₁𝑞,(𝑡,=𝑦−(𝑡),2𝑦𝑣̇(₂𝑡),𝑣(𝑡),𝑞)𝑘(𝑡)+𝑦(𝑡)+𝑦²(𝑡)+𝑣²(𝑡)+1₁𝑞,(𝑡,=𝑦(𝑦𝑡₂),𝑦̇(𝑡),𝑣(𝑡),𝑞)𝑙(𝑡)+𝑣²(𝑡)𝑞,₁(𝑡,=𝑦(4𝑡)𝑦,₂𝑦̇(𝑡),𝑣(𝑡),𝑞)(𝑡)+5𝑣²(𝑡)+𝑦²(𝑡)+1𝑞.(44) The Mond-Weir higher-order dual of the control problem is subjemctatxo2𝑦^𝑏𝑎₂𝑦²(𝑡)+𝑣²(𝑡)𝑑𝑡,(45)6𝑦₂(𝑡)+𝑣²(𝑡)≥0,(46)𝑦(𝑎)(𝑡=)+𝛾₁9𝑣²(𝑡)≥0,(47)𝑣(𝑎)=𝛿₁,𝑦(𝑏)=𝛾₂,𝛼(𝑡)=𝛽(,𝑡)=𝛼𝑣₁(𝑏)=𝛿₂,(48)(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 𝛼∶𝐼→ℝ^𝑟, 𝛽∶𝐼→ℝ^𝑠, let (i)ℎ(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=ℎ₁(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=0,(ii)𝑘(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=𝑘₁(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=0,(iii)𝑙(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=𝑙₁(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=0,(iv)∇_𝑝ℎ(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=𝑓_𝑥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))−(𝑑/𝑑𝑡)𝑓_̇𝑥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡)),(v)∇_𝑝𝛼(𝑡)^𝑇𝑘(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0) = 𝑔_𝑥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))^𝑇𝛼(𝑡)−(𝑑/𝑑𝑡)𝑔_̇𝑥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))^𝑇𝛼(𝑡),(vi)∇_𝑝𝛽(𝑡)^𝑇𝑙(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0) = 𝐺_𝑥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))^𝑇𝛽(𝑡)−(𝑑/𝑑𝑡)𝐺_̇𝑥(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))^𝑇𝛽(𝑡),(vii)∇_𝑞ℎ₁(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0)=𝑓_𝑢(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡)),(viii)∇_𝑞𝛼(𝑡)^𝑇𝑘₁(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0) = 𝑔_𝑢(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡))^𝑇𝛼(𝑡),(ix)∇_𝑞𝛽(𝑡)^𝑇𝑙₁(𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),0) = 𝐺_𝑢(𝑡,𝑥(𝑡),𝑥̇(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+𝜌₂≥0. Moreover, if ℎ𝑡,−𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝+ℎ₁𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞−𝑝^𝑇∇_𝑝ℎ𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑝𝑞^𝑇∇_𝑞ℎ₁𝑡,𝑥(𝑡),𝑥̇(𝑡),𝑢(𝑡),𝑞≥0,(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.