Higher-Order Generalized Invexity in Control Problems

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.


Introduction
We consider the control problem (CP) min b a f (t, x(t), ẋ(t), u(t))dt, ( 1 ) subject to g(t, x(t), ẋ(t), u(t)) ≤ 0, G(t, x(t), ẋ(t), u(t)) = 0, x(a) = γ 1 , x(b) = γ 2 ; where f , g, and G are twice continuously differentiable functions from 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.Higherorder 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.

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 by where κ is a given boundary value; thus, d/dt = D except at discontinuities.The higher-order generalized invexity functions are defined as follows.
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 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, μ 0 is nonnegative constant, β(t) is continuous in a ≤ t ≤ b, and μ 0 , 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 μ 0 can be taken equal to 1.
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.
Proof.The proof is similar to that of Theorem 3.4.
Proof.The proof is similar to that of Theorem 3.5.

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.