# Higher-Order Generalized Invexity in Control Problems

Authors: S. K. Padhan; C. Nahak
Journal: Journal of Control Science and Engineering
 (2011)

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## 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.


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## Body

## 1. Introduction

We consider the control problem(1)(CP)min⁡∫abf(t,x(t),ẋ(t),u(t))dt,(2)subjecttog(t,x(t),ẋ(t),u(t))≤0,(3)G(t,x(t),ẋ(t),u(t))=0,(4)x(a)=γ1,x(b)=γ2;u(a)=δ1,u(b)=δ2,
where f, g, and G are twice continuously differentiable functions from I×ℝn×ℝn×ℝm into ℝ, ℝr, and ℝs, 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

LetS(I,ℝn) denote the space of piecewise smooth functions x with norm ∥x∥=∥x∥∞+∥Dx∥∞, where the differentiation operator D is given by(5)u=Dx⟺x(t)=κ+∫atu(s)ds,
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 functionalF(x,u)=∫abf(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×ℝn×ℝn×ℝn×ℝn×ℝm×ℝm→ℝn,ξ:I×ℝn×ℝn×ℝn×ℝn×ℝm×ℝm→ℝm, h:I×ℝn×ℝn×ℝm×ℝn→ℝ,h1:I×ℝn×ℝn×ℝm×ℝm→ℝ and ρ∈ℝ, such that
(6)F(x,u)-F(y,v)≥∫ab[η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇ph(t,y(t),ẏ(t),v(t),p)+ξ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇qh1(t,y(t),ẏ(t),v(t),q)+h(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)-pT∇ph(t,y(t),ẏ(t),v(t),p)-qT∇qh1(t,y(t),ẏ(t),v(t),q)+ρ‖θ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))‖2]dt.Definition 2.2.
The scalar functionalF(x,u)=∫abf(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×ℝn×ℝn×ℝn×ℝn×ℝm×ℝm→ℝn,ξ:I×ℝn×ℝn×ℝn×ℝn×ℝm×ℝm→ℝm, h:I×ℝn×ℝn×ℝm×ℝn→ℝ,h1:I×ℝn×ℝn×ℝm×ℝm→ℝ and ρ∈ℝ, such that
(7)∫ab[η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇ph(t,y(t),ẏ(t),v(t),p)+ξ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇qh1(t,y(t),ẏ(t),v(t),q)+ρ‖θ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))‖2]dt≥0⟹F(x,u)-F(y,v)-∫ab[qT∇qh(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)-pT∇ph(t,y(t),ẏ(t),v(t),p)-qT∇qh1(t,y(t),ẏ(t),v(t),q)]dt≥0.Definition 2.3.
The scalar functionalF(x,u)=∫abf(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×ℝn×ℝn×ℝn×ℝn×ℝm×ℝm→ℝn, ξ:I×ℝn×ℝn×ℝn×ℝn×ℝm×ℝm→ℝm, h:I×ℝn×ℝn×ℝm×ℝn→ℝ, h1:I×ℝn×ℝn×ℝm×ℝm→ℝ and ρ∈ℝ, such that
(8)F(x,u)-F(y,v)-∫ab[qT∇qh(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)-pT∇ph(t,y(t),ẏ(t),v(t),p)-qT∇qh1(t,y(t),ẏ(t),v(t),q)]dt≤0⟹∫ab[η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇ph(t,y(t),ẏ(t),v(t),p)+ξ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇qh1(t,y(t),ẏ(t),v(t),q)+ρ‖θ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))‖2]dt≤0.Remark 2.4.
If(9)h(t,y(t),ẏ(t),v(t),p)=pT[fx(t,y(t),ẏ(t),v(t))-ddtfẋ(t,y(t),ẏ(t),v(t))],h1(t,y(t),ẏ(t),v(t),q)=qTfu(t,y(t),ẏ(t),v(t)),
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):(10)(MHCD)max⁡∫ab[β1(t)Tf(t,y(t),ẏ(t),v(t))+α(t)Tg(t,y(t),ẏ(t),v(t))+β(t)TG(t,y(t),ẏ(t),v(t))+h(t,y(t),ẏ(t),v(t),p)+α(t)Tk(t,y(t),ẏ(t),v(t),p)+β(t)Tl(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),p)+α1(t)Tk1(t,y(t),ẏ(t),v(t),q)+β1(t)Tl1(t,y(t),ẏ(t),v(t),q)]dt,(11)subjectto∇ph(t,y(t),ẏ(t),v(t),p)+∇pα(t)Tk(t,y(t),ẏ(t),v(t),p)+∇pβ(t)Tl(t,y(t),ẏ(t),v(t),p)=0,(12)∇qh1(t,y(t),ẏ(t),v(t),q)+∇qα1(t)Tk1(t,y(t),ẏ(t),v(t),q)+∇qβ1(t)Tl1(t,y(t),ẏ(t),v(t),q)=0,(13)y(a)=γ1,y(b)=γ2;v(a)=δ1,v(b)=δ2,(14)α(t)∈R+r,β(t)∈Rs,α1(t)∈R+r1,β1(t)∈Rs1,p∈Rn,q∈Rm.Remark 3.1.
If(15)h(t,y(t),ẏ(t),v(t),p)=p[fx(t,y(t),ẏ(t),v(t))-ddtfẋ(t,y(t),ẏ(t),v(t))]+12p[d2dt2fxx(t,y(t),ẏ(t),v(t))-2ddtfxẋ(t,y(t),ẏ(t),v(t))+d2dt2fẋẋ(t,y(t),ẏ(t),v(t))]p,h1(t,y(t),ẏ(t),v(t),q)=q[fu(t,y(t),ẏ(t),v(t))]+12q[fuu(t,y(t),ẏ(t),v(t))+2fxu(t,y(t),ẏ(t),v(t))]q,kj(t,y(t),ẏ(t),v(t),p)=p[gxj(t,y(t),ẏ(t),v(t))-ddtgẋj(t,y(t),ẏ(t),v(t))]+12p[d2dt2gxxj(t,y(t),ẏ(t),v(t))-2ddtgxẋj(t,y(t),ẏ(t),v(t))+d2dt2gẋẋj(t,y(t),ẏ(t),v(t))]p,j=1,2,3,…,r,k1j(t,y(t),ẏ(t),v(t),q)=qguj(t,y(t),ẏ(t),v(t))+12q[guuj(t,y(t),ẏ(t),v(t))+2gxuj(t,y(t),ẏ(t),v(t))]q,j=1,2,…,r1,li(t,y(t),ẏ(t),v(t),p)=p[Gxi(t,y(t),ẏ(t),v(t))-ddtGẋi(t,y(t),ẏ(t),v(t))]+12p[d2dt2Gxxi(t,y(t),ẏ(t),v(t))-2ddtGxẋi(t,y(t),ẏ(t),v(t))+d2dt2Gẋẋi(t,y(t),ẏ(t),v(t))]p,i=1,2,3,…,s,l1i(t,y(t),ẏ(t),v(t),q)=qGui(t,y(t),ẏ(t),v(t))+12q[Guui(t,y(t),ẏ(t),v(t))+2Gxui(t,y(t),ẏ(t),v(t))]q,i=1,2,3,…,s1,
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 ∫abf(t,·,·,·)dt, ∫abα(t)Tg(t,·,·,·)dt and ∫abβ(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):
(16)∫abf(t,x(t),ẋ(t),u(t))dt≥∫ab[f(t,y(t),ẏ(t),v(t))+α(t)Tg(t,y(t),ẏ(t),v(t))+β(t)TG(t,y(t),ẏ(t),v(t))+h(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)+α(t)Tk(t,y(t),ẏ(t),v(t),p)+α1(t)Tk1(t,y(t),ẏ(t),v(t),q)+β(t)Tl(t,y(t),ẏ(t),v(t),p)+β1(t)Tl1(t,y(t),ẏ(t),v(t),q)]dt.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
(17)∫abf(t,x(t),ẋ(t),u(t))dt-∫ab[f(t,y(t),ẏ(t),v(t))+α(t)Tg(t,y(t),ẏ(t),v(t))+β(t)TG(t,y(t),ẏ(t),v(t))+h(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)+α(t)Tk(t,y(t),ẏ(t),v(t),p)+α1(t)Tk1(t,y(t),ẏ(t),v(t),q)+β(t)Tl(t,y(t),ẏ(t),v(t),p)+β1(t)Tl1(t,y(t),ẏ(t),v(t),q)]dt≥[∫abf(t,x(t),ẋ(t),u(t))dt-∫abf(t,y(t),ẏ(t),v(t))dt]+[∫abα(t)Tg(t,x(t),ẋ(t),u(t))dt-∫abα(t)Tg(t,y(t),ẏ(t),v(t))dt]+[∫abβ(t)TG(t,x(t),ẋ(t),u(t))dt-∫abβ(t)TG(t,y(t),ẏ(t),v(t))dt]-∫ab[(t)Th(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)+α(t)Tk(t,y(t),ẏ(t),v(t),p)+α1(t)Tk1(t,y(t),ẏ(t),v(t),q)+β(t)Tl(t,y(t),ẏ(t),v(t),p)+β1(t)Tl1(t,y(t),ẏ(t),v(t),q)]dt≥0.
(by the higher-order ρ-(η,ξ,θ)-invexity of ∫abf(t,·,·,·)dt,∫abα(t)Tg(t,·,·,·)dt) and ∫abβ(t)TG(t,·,·,·)dt,ρ0+ρ1+ρ2≥0 and equations (11)–(14)). This completes the proof.We construct the following example which verifies Theorem3.2 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:(18)min⁡∫ab(x2(t)-x(t)-u(t))dt,(19)subjecttou(t)-x2(t)≤0,x(t)-u2(t)=0,x(a)=γ1,x(b)=γ2,u(a)=δ1,u(b)=δ2,
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:
(20)η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))=-(x2(t)+x(t)+u(t)+y(t)),ξ(t,x(t),y(t),ẋ(t),ẏ(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),ẋ(t),ẏ(t),u(t),v(t))=x2(t)+y2(t),h(t,y(t),ẏ(t),v(t),p)=(2y(t)-1)p-y2(t)-v2(t),k(t,y(t),ẏ(t),v(t),p)=-2y(t)p-v2(t),l(t,y(t),ẏ(t),v(t),p)=p-y2(t)-v2(t),h1(t,y(t),ẏ(t),v(t),q)=-q-y2(t),k1(t,y(t),ẏ(t),v(t),q)=q-v2(t),l1(t,y(t),ẏ(t),v(t),q)=-y2(t)-v2(t).
The Mangasarian higher-order dual of the control problem is
(21)max⁡∫ab(-4y2(t)-6v2(t))dt,(22)subjecttoy(a)=γ1,y(b)=γ2;v(a)=δ1,v(b)=δ2,α(t)=β(t)=α1(t)=β1(t)=1.
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), then(23)μ0fx(t,y(t),ẏ(t),v(t))+gx(t,y(t),ẏ(t),v(t))Tα(t)+Gx(t,y(t),ẏ(t),v(t))Tβ(t)=ddt((t,y(t),ẏ(t),v(t))Tμ0fẋ(t,y(t),ẏ(t),v(t))+gẋ(t,y(t),ẏ(t),v(t))Tα(t)+Gẋ(t,y(t),ẏ(t),v(t))Tβ(t)),(24)μ0fu(t,y(t),ẏ(t),v(t))+gu(t,y(t),ẏ(t),v(t))Tα(t)+hu(t,y(t),ẏ(t),v(t))Tβ(t)=0,(25)α(t)Tg(t,y(t),ẏ(t),v(t))=0,(26)β(t)TG(t,y(t),ẏ(t),v(t))=0,(27)α(t)∈R+r
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→ℝr, β¯:I→ℝs, 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)=fx(t,x¯(t),x¯(t),u¯(t))-(d/dt)fẋ(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)gẋ(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)Gẋ(t,x¯(t),x¯(t),u¯(t))Tβ¯(t),(vii)
∇qh1(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 ∫abf(t,·,·,·)dt, ∫abα¯(t)Tg(t,·,·,·)dt and ∫abβ¯(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+ρ2≥0. Moreover, if
(28)α¯(t)Tg(t,x¯(t),x¯(t),u¯(t))+β¯(t)TG(t,x¯(t),x¯(t),u¯(t))×h(t,x¯(t),x¯̇(t),u¯(t),p¯)+h1(t,x¯(t),x¯̇(t),u¯(t),q¯)+α¯k(t,x¯(t),x¯̇(t),u¯(t),p¯)+α¯1k1(t,x¯(t),x¯̇(t),u¯(t),q¯)+β¯l(t,x¯(t),x¯̇(t),u¯(t),p¯)+β¯1l1(t,x¯(t),x¯̇(t),u¯(t),q¯)≥0,
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
(29)∫abf(t,x(t),ẋ(t),u(t))dt<∫abf(t,x¯(t),x¯̇(t),u¯(t))dt.
Since∫abf(t,·,·,·)dt, ∫abα¯(t)Tg(t,·,·,·)dt and ∫abβ¯(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
(30)∫abf(t,x(t),ẋ(t),u(t))dt-∫abf(t,x¯(t),x¯̇(t),u¯(t))dt≥∫ab[α¯(t)Tg(t,x¯(t),x¯̇(t),u¯(t))+β¯(t)TG(t,x¯(t),x¯̇(t),u¯(t))+h(t,x¯(t),x¯̇(t),u¯(t),p¯)+h1(t,x¯(t),x¯̇(t),u¯(t),q¯)+α¯k(t,x¯(t),x¯̇(t),u¯(t),p¯)+α¯1k1(t,x¯(t),x¯̇(t),u¯(t),q¯)+β¯l(t,x¯(t),x¯̇(t),u¯(t),p¯)+β¯1l1(t,x¯(t),x¯̇(t),u¯(t),q¯)]dt(byequations(2),(3),(11),(12),and(14))≥0(byequations(28)).
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):(31)(MWHCD)max⁡∫ab[qT∇qf(t,y(t),ẏ(t),v(t))+h(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)-pT∇ph(t,y(t),ẏ(t),v(t),p)-qT∇qh1(t,y(t),ẏ(t),v(t),q)]dt,(32)subjectto∇ph(t,y(t),ẏ(t),v(t),p)+∇pα(t)Tk(t,y(t),ẏ(t),v(t),p)+∇pβ(t)Tl(t,y(t),ẏ(t),v(t),p)=0,(33)∇qh1(t,y(t),ẏ(t),v(t),q)+∇qα1(t)Tk1(t,y(t),ẏ(t),v(t),q)+∇qβ1(t)Tl1(t,y(t),ẏ(t),v(t),q)=0,(34)α(t)Tg(t,y(t),ẏ(t),v(t))+α(t)Tk(t,y(t),ẏ(t),v(t),p)+α1(t)Tk1(t,y(t),ẏ(t),v(t),q)-pT∇pα(t)Tk(t,y(t),ẏ(t),v(t),p)-qT∇qα1(t)Tk1(t,y(t),ẏ(t),v(t),q)≥0,(35)β1(t)TG(t,y(t),ẏ(t),v(t))+β(t)Tl(t,y(t),ẏ(t),v(t),p)+β1(t)Tl1(t,y(t),ẏ(t),v(t),q)-pT∇pβ(t)Tl(t,y(t),ẏ(t),v(t),p)-qT∇qβ1(t)Tl1(t,y(t),ẏ(t),v(t),q)≥0,(36)y(a)=γ1,y(b)=γ2,v(a)=δ1,v(b)=δ2,(37)α(t)∈R+r,β(t)∈Rs,α1(t)∈R+r1,β1(t)∈Rs1,p∈Rn,q∈Rm.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 ∫abf(t,·,·,·)dt, ∫abα(t)Tg(t,·,·,·)dt and ∫abβ(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),
(38)∫abf(t,x(t),ẋ(t),u(t))dt≥∫ab[qT∇qf(t,y(t),ẏ(t),v(t))+h(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)-pT∇ph(t,y(t),ẏ(t),v(t),p)-qT∇qh1(t,y(t),ẏ(t),v(t),q)]dt.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
(39)α(t)Tg(t,x(t),ẋ(t),u(t))-α(t)Tg(t,y(t),ẏ(t),v(t))-α(t)Tk(t,y(t),ẏ(t),v(t),p)-α1(t)Tk1(t,y(t),ẏ(t),v(t),q)+pT∇pα(t)Tk(t,y(t),ẏ(t),v(t),p)+qT∇qα1(t)Tk1(t,y(t),ẏ(t),v(t),q)≤0,(40)β1(t)TG(t,x(t),ẋ(t),u(t))-β1(t)TG(t,y(t),ẏ(t),v(t))-β(t)Tl(t,y(t),ẏ(t),v(t),p)-β1(t)Tl1(t,y(t),ẏ(t),v(t),q)+pT∇pβ(t)Tl(t,y(t),ẏ(t),v(t),p)+qT∇qβ1(t)Tl1(t,y(t),ẏ(t),v(t),q)≤0.
Since ∫abα(t)Tg(t,·,·,·)dt and ∫abβ(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
(41)∫ab[η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T(∇pα(t)Tk(t,y(t),ẏ(t),v(t),p)+∇pβ(t)Tl(t,y(t),ẏ(t),v(t),p))+ξ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T+(∇qα1(t)Tk1(t,y(t),ẏ(t),v(t),q)+∇qβ1(t)Tl1(t,y(t),ẏ(t),v(t),q))+(ρ1+ρ2)‖θ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))‖2]dt≤0⟹∫ab[η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇ph(t,y(t),ẏ(t),v(t),p)+ξ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))T∇qh1(t,y(t),ẏ(t),v(t),q)+ρ‖θ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))‖2]dt≥0(byequations(32),(33)andρ0+ρ1+ρ2≥0)⟹∫abf(t,x(t),ẋ(t),u(t))dt-∫ab[qT∇qf(t,y(t),ẏ(t),v(t))+h(t,y(t),ẏ(t),v(t),p)+h1(t,y(t),ẏ(t),v(t),q)-pT∇ph(t,y(t),ẏ(t),v(t),p)-qT∇qh1(t,y(t),ẏ(t),v(t),q)]dt≥0.
by the higher-order  ρ0-(η,ξ,θ)-pseudoinvexity of ∫abf(t,·,·,·)dt. This completes the proof.We construct the following example which verifies Theorem4.1 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:(42)min⁡∫ab(x2(t)+u2(t)+x(t)+u(t))dt,(43)subjecttox2(t)-x(t)u(t)≤0,u2(t)-x(t)=0,x(a)=γ1,x(b)=γ2,u(a)=δ1,u(b)=δ2,
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:
(44)η(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))=-(x2(t)+u(t)+1),ξ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))=-(u2(t)+x(t)+1),θ(t,x(t),y(t),ẋ(t),ẏ(t),u(t),v(t))=x2(t)y2(t)+u(t)y2(t)+u2(t)v2(t)+x(t)v2(t),h(t,y(t),ẏ(t),v(t),p)=-(y2(t)+v2(t)-(y2(t)+v2(t)+1)p),k(t,y(t),ẏ(t),v(t),p)=y(t)v(t)+v2(t)+y2(t)p,h1(t,y(t),ẏ(t),v(t),q)=-(2v2(t)+y(t)+(y2(t)+v2(t)+1)q),h1(t,y(t),ẏ(t),v(t),q)=-(2v2(t)+y(t)+(y2(t)+v2(t)+1)q),k1(t,y(t),ẏ(t),v(t),q)=y2(t)+v2(t)q,l1(t,y(t),ẏ(t),v(t),q)=4y2(t)+5v2(t)+(y2(t)+1)q.
The Mond-Weir higher-order dual of the control problem is
(45)max⁡∫ab(y2(t)+v2(t))dt,(46)subjectto2y2(t)+v2(t)≥0,(47)6y2(t)+9v2(t)≥0,(48)y(a)=γ1,y(b)=γ2,v(a)=δ1,v(b)=δ2,(49)α(t)=β(t)=α1(t)=β1(t)=1.
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→ℝr, β¯:I→ℝs, 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)=fx(t,x¯(t),x¯̇(t),u¯(t))-(d/dt)fẋ(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)gẋ(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)Gẋ(t,x¯(t),x¯̇(t),u¯(t))Tβ¯(t),(vii)
∇qh1(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 Theorem3.4.Theorem 4.4 (converse duality).
Let(y¯(t),v¯(t),α¯(t),β¯(t),p¯,q¯) be an optimal solution of (MWHCD). Suppose that ∫abf(t,·,·,·)dt, ∫abα¯(t)Tg(t,·,·,·)dt and ∫abβ¯(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
(50)h(t,x¯(t),x¯̇(t),u¯(t),p¯)+h1(t,x¯(t),x¯̇(t),u¯(t),q¯)-p¯T∇p¯h(t,x¯(t),x¯̇(t),u¯(t),p¯)-q¯T∇q¯h1(t,x¯(t),x¯̇(t),u¯(t),q¯)≥0,
then (x¯(t),u¯(t)) is an optimal solution of (CP).Proof.
The proof is similar to that of Theorem3.5.


## 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. Example3.3 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.



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*Source: 127208-2011-05-26.xml*