Sufficiency and mixed type duality for multiobjective variational control problems involving α -V-univexity

In this paper, we focus our study on a multiobjective variational control problem and establish sufficient optimality conditions under the assumptions of α -V-univex function. Furthermore, mixed type duality results are also discussed under the aforesaid assumption in order to relate the primal and dual problems. Examples are given to show the existence of α -V-univex function and to elucidate duality result.


1.
Introduction. Often, in real-world problems, we come across situations where we have to optimize more than one objective functions which are conflicting in nature. These types of problems can be solved by using multiobjective optimization techniques.
The calculus of variation plays a significant role in many areas of pure and applied mathematics. It is concerned with optimization of functionals, which are mappings from a set of functions to the real numbers and are often expressed as definite integrals involving functions and their derivatives. This technique was developed to solve the problems of finding the best possible objects, for example, to find the shortest distance between two points, minimal surfaces or trajectory of the fastest travel.
Thereafter, control theory emerged as an extension of the calculus of variation. The control problem is to optimize a cost functional that is a function of state and control variables. It finds applications in diverse areas of science, engineering and decision making. Hanson [7,8] was the first to observe the relationship between calculus of variation, control problems and mathematical programming and obtained duality results for control problems under the assumption of convexity. Moreover, the most systematic applications of duality to optimal control theory were developed by Rockafellar [20,21]. Thereafter, several researchers have been interested in the optimality conditions and duality theorems for variational control problems, see for example, [1, 2, 6, 10-12, 15-17, 22]. For details in control theory, one can refer to the book of Craven [4].
As we know, convexity is one of the most important hypothesis in optimization theory but it is not sufficient for many real world mathematical models. Thus, Mond and Smart [15] extended the work of Mond and Hanson [9], under the assumptions of invexity. Liang et al. [13,14] defined a new class of functions, known as (F, α, ρ, d)-convex functions, which unifies several concepts of generalized convexity. Various new classes of functions were introduced by several authors, see [2, 10-12, 16-19, 22, 23], on account of certain limitations of convexity. Bhatia and Kumar [5] introduced multiobjective control problem and established duality results under generalized ρ-invexity assumptions. Thereafter, Gulati et al. [6] obtained Fritz John and Kuhn-Tucker type necessary optimality conditions and duality conditions for multiobjective control problems under generalized invexity.
Bector et al. [3] introduced the concept of univex functions as a generalization of invex functions. Xu [23] introduced two mixed type duals for multiobjective programming and multiobjective fractional programming, respectively and derived duality results under the assumptions of generalized (F, ρ)-convexity. Later, Ahmad and Gulati [1] formulated mixed type duality for multiobjective variational problems under the assumption of generalized (F, ρ)-convexity. In the inspiration of the above authors, we obtain sufficient optimality conditions and duality results for mixed type dual of a multiobjective variational control problem under the assumption of α-V-univexity.
This paper is organized as follows: In Section 2, we recall some preliminary definitions and introduce the concept of α-V-univexity. In Section 3, we derive the sufficient optimality conditions for multiobjective variational control problems under the assumption of α-V-univex functions and finally in Section 4, we prove duality results for mixed type multiobjective dual problem.
2. Notations and preliminaries. Let R n denote the n-dimensional Euclidean space. Let y, z ∈ R n , we denote: y z ⇔ y i z i , i = 1, 2, . . . , n; y ≤ z ⇔ y z and y = z; y < z ⇔ y i < z i , i = 1, 2, . . . , n. Let . . , n} be continuously differentiable functions. Consider the function f (t, x(t),ẋ(t), u(t),u(t)), where t is the independent variable, x : I → R n is the state variable and u : I → R m is the control variable. u(t) is related to x(t) via the state equation h(t, x(t),ẋ(t), u(t),u(t)) = 0, where the dot denotes the derivative with respect to t. f ix , f iẋ , f iu and f iu denote the partial derivatives of f i with respect to x,ẋ, u andu, respectively. For instance, Similarly, g jx , g jẋ , g ju , g ju and h kx , h kẋ , h ku , h ku can be defined. For notational convenience, we use x,ẋ, u,u in place of x(t),ẋ(t), u(t),u(t), respectively. Let X denote the space of all piecewise smooth functions x : I → R n with norm x = x ∞ + Dx ∞ and Y denote the space of all piecewise smooth functions u : I → R m with the norm u ∞ , where the differentiation operator D is given by where γ is a given boundary value. Therefore D = d/dt except at discontinuities.
We consider the following multiobjective variational control problem: x,ẋ, u,u) 0, h(t, x,ẋ, u,u) = 0} denote the set of all feasible solutions to (CP).
x,ẋ,ū,u)dt for some i ∈ P and b a f r (t, x,ẋ, u,u)dt b a f r (t,x,ẋ,ū,u)dt for all r ∈ P \ {i}.
In the case of maximization, the signs of above inequalities are reversed. It follows that if (x,ū) ∈ S is efficient for (CP), then it is also weakly efficient for (CP).
Example 2.1. Let I = [0, 1] and let X and Y denote the space of all piecewise smooth functions x : I → R + and u : I → R + respectively. Let ψ : and take b • = 3. Then, Therefore, it follows that Hence, it follows that 3. Sufficient optimality conditions. In this section, we establish the following sufficient optimality conditions involving α-V-univexity assumptions.
This verifies the weak duality theorem.
Let us now turn our attention towards strong duality theorem. For this, we shall consider the following Proposition as an extension to the Proposition 1 given in Ahmad and Gulati [1].
Proof. Since (x • , u • ) is an efficient solution for (CP) and every efficient solution for (CP) is also weakly efficient. Therefore by Proposition 4.1, there exists λ ∈ R p and piecewise smooth functions µ : I → R m and ν : I → R n satisfying (26) to (30).