On Higher-Order PDE Constrained Multiobjective Optimization Models

Abstract

In this paper, we formulate and prove necessary conditions of efficiency for a new class of multiobjective variational models governed by higher order partial derivatives. More precisely, we consider a multiobjective optimization model of minimizing a vector of multiple integral functionals subject to certain higher order differential equations and/or inequations. The main results are derived by applying suitable techniques coming from variational calculus. The current contribution lies in vector-valued functionals given by multiple integrals, constraint coupling, and the characterization of efficiency criteria.

1. Framework and Problem Description

Over time, several authors have been interested in the investigation of vector (multiobjective) variational problems by considering generalized convexity (see, for instance, Mititelu and Udrişte [1], Mititelu [2], Nahak and Mohapatra [3], and Preda and Gramatovici [4]). Moreover, many studies extend this notion of convexity and develop a multitime variational theory, sometimes using a geometrical language (see, for instance, Treanţă [5,6,7], Vetro and Zeng [8]). For other ideas related to this topic, we address the readers to Chinchuluun and Pardalos [9], Geoffrion [10], Jagannathan [11], Treanţă [12], and Cen et al. [13,14,15]. Sadek et al. [16,17] extended the classical framework to conformal and trigonometric fractional calculus and provided new properties and applications. Moreover, the Galerkin Bell method to solve the fractional optimal control problems with inequality constraints, and the extended block Arnoldi method to solve generalized differential Sylvester equations have been considered in recent studies (see Sadek et al. [18,19]).

In this work, we extend and further develop some optimization outcomes in connection with the efficiency of admissible solutions associated with a family of multiple objective non-fractional programming problems implying higher order partial derivatives. Thus, we introduce and formulate a study on multiobjective (vector) variational model consisting in extremizing a vector made up of multiple integral functionals constrained by partial differential equations and inequations of higher order. The main novelty element associated with this paper is the presence of higher order partial derivatives. Also, as a natural consequence, the corresponding context is dominated by new and more complex techniques and methods. Moreover, the current paper is motivated by various applications in wide areas of research and in natural phenomena, where partial derivatives (of higher order) are requested.

The proper motivation for investigating such issues is, more precisely, the following:

(i) Pointwise state-constrained optimization problem:

$${\displaystyle \underset{u,\vartheta }{min}\frac{1}{2}{\int }_{{\Omega }_{t_0,t_1}}{\left(u\left(t\right)-sin\left(2\pi t^1{t}^2\right)\right)}^{2}d{t}^{1}d{t}^2+\frac{\alpha }{2}{\int }_{{\Omega }_{t_0,t_1}}{\vartheta }^2\left(t\right)d{t}^{1}d{t}^2}$$

subject to

$${\displaystyle -\Delta u\left(t\right)=\vartheta \left(t\right),t\in {\Omega }_{t_0,t_1};u\left(t\right)=0,t\in \partial {\Omega }_{t_0,t_1},}$$

studied by Udrişte and Matei [20], as well, by considering a simplified multi-time maximum principle.

(ii) Neumann boundary control:

$${\displaystyle J\left(\vartheta (·)\right)=\frac{1}{2}{\int }_{\Omega }{(x\left(t\right)-z_d)}^{2}d{t}^1\cdots d{t}^m+\frac{\beta }{2}{\int }_{\Gamma }{\vartheta }^2\left(t\right)ds,}$$

where ${\displaystyle \Omega }$ is a bounded set in ${\displaystyle R^m}$ having the boundary ${\displaystyle \Gamma }$ of ${\displaystyle C^2}$-class, the pair ${\displaystyle (x,\vartheta )}$ satisfies ${\displaystyle -\Delta x+x=f}$ in ${\displaystyle \Omega }$ and ${\displaystyle \frac{\partial x}{\partial n}=\vartheta }$ on ${\displaystyle \Gamma }$. The above-mentioned function f is a source term, and the function $\vartheta$ is a control variable. Since ${\displaystyle \frac{\beta }{2}{\int }_{\Gamma }{\vartheta }^2\left(t\right)ds}$ (with ${\displaystyle \beta >0}$) is proportional with the consumed energy, the extremizing of the functional ${\displaystyle J}$ is a compromise between finding $\vartheta$ such that x is close to the desired profile $z_d$ and the energy consumption. The control variable is named boundary control because it acts on the boundary ${\displaystyle \Gamma }$.

The above optimization problems (implying second-order partial derivatives) were taken from the specialized literature. Other illustrative examples and new points of view, can be read in Udrişte and Matei [20] and Raymond [21].

The passing from the first-order partial derivatives to the higher-order partial derivatives is a very complex task because it requests specific techniques and an appropriate mathematical framework. Studies regarding higher-order variational calculus and optimal control theory have been formulated until now by certain researchers. However, the current contribution lies in vector-valued functionals given by multiple integrals, constraint coupling, and the characterization of efficiency criteria.

Before presenting our results, for the completeness of the exposition, we set the following notations.

Consider the real multi-interval ${\displaystyle \Omega :=[t_0,t_1]\subseteq {\mathbb{R}}^w}$ (hyper-parallelepiped determined by the diagonal corners $t_0=(t_0^1,\cdots t_0^w)$ and $t_1:=(t_1^1,\cdots t_1^w)$) and

$${\displaystyle \sigma =\left({\sigma }_{\xi }\right):\Omega \times {\mathbb{R}}^{n(\mu +1)}\to {\mathbb{R}}^a,\xi =\overline{1,a},}$$

$${\displaystyle \sigma =({\sigma }_1(t,\pi \left(t\right),{\pi }_{{\gamma }_1}\left(t\right),\dots ,{\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}\left(t\right)),}$$

$${\displaystyle \dots ,{\sigma }_a(t,\pi \left(t\right),{\pi }_{{\gamma }_1}\left(t\right),\dots ,{\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}\left(t\right))),}$$

a ${\displaystyle C^{\mu +1}}$-class functional, where ${\displaystyle {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}\left(t\right):=\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\pi \left(t\right)}$, with ${\displaystyle \mu \ge 1}$ a fixed natural number, and ${\gamma }_1,\dots ,{\gamma }_{\mu }\in \{1,2,\dots ,w\}$. Also, let ${\displaystyle \psi =\left({\psi }_1,\dots ,{\psi }_k\right):\Omega \times {\mathbb{R}}^{n(\mu +1)}\to {\mathbb{R}}^k}$, with ${\displaystyle k<n}$, and ${\displaystyle \tau =\left({\tau }_1,\dots ,{\tau }_r\right):\Omega \times {\mathbb{R}}^{n(\mu +1)}\to {\mathbb{R}}^r}$, with ${\displaystyle r<n}$, two ${\displaystyle C^{\mu +1}}$-class functionals. Assume that the previous ${\displaystyle C^{\mu +1}}$-class Lagrangians,

$${\displaystyle {\sigma }_{\xi }(t,\pi \left(t\right),{\pi }_{{\gamma }_1}\left(t\right),\dots ,{\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}\left(t\right)),\xi =\overline{1,a},}$$

generate the multiple integral functionals

$${\displaystyle F_{\xi }\left(\pi (·)\right):={\int }_{\Omega }{\sigma }_{\xi }(t,\pi \left(t\right),{\pi }_{{\gamma }_1}\left(t\right),\dots ,{\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}\left(t\right))dv,}$$

$${\displaystyle \xi =\overline{1,a},}$$

where $dv:=d{t}^1\cdots d{t}^w$ (volume element).

Let ${\displaystyle C^{\infty }\left(\Omega ,{\mathbb{R}}^n\right)}$ be the space of all functions ${\displaystyle \pi :\Omega \to {\mathbb{R}}^n}$ of ${\displaystyle C^{\infty }}$-class, with the norm

$${\displaystyle \parallel \pi \parallel :={\parallel \pi \parallel }_{\infty }+\sum _{\beta =1}^{\mu }{\parallel {\pi }_{{\gamma }_1\cdots {\gamma }_{\beta }}\parallel }_{\infty },}$$

or, for generality, we can assume the Sobolev-type spaces for our framework.

As usual, two vectors, ${\displaystyle b=\left(b_1,\dots ,b_s\right),v=\left(v_1,\dots ,v_s\right)}$ in ${\displaystyle {\mathbb{R}}^s}$ are related via

$${\displaystyle b=v\iff b_i=v_i,b\le v\iff b_i\le v_i}$$

$${\displaystyle b<v\iff b_i<v_i,b⪯v\iff b\le v,b\ne v,i=\overline{1,s}.}$$

Also, we underline that the argument of our Lagrangians is denoted as follows

$${\displaystyle z_{\pi }\left(t\right):=(t,\pi \left(t\right),{\pi }_{{\gamma }_1}\left(t\right),\dots ,{\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}\left(t\right)).}$$

Using these ingredients, we formulate the multiobjective variational problem (MVP) (higher-order PDE constrained optimization problem),

$$\begin{gathered} {\displaystyle \underset{\pi \left(·\right)}{min}Q\left(\pi (·)\right)=\left({\int }_{\Omega }{\sigma }_1\left(z_{\pi }\left(t\right)\right)dv,\dots ,{\int }_{\Omega }{\sigma }_a\left(z_{\pi }\left(t\right)\right)dv\right) \\ \mathrm{subject}\mathrm{to}\pi \left(·\right)\in \text{Q}\left(\Omega \right),} \end{gathered}$$

where the set (domain) ${\displaystyle \text{Q}\left(\Omega \right)}$ of all feasible solutions is

$${\displaystyle \pi \in C^{\infty }\left(\Omega ,{\mathbb{R}}^n\right),\pi \left(t_{\epsilon }\right)={\pi }_{\epsilon },{\pi }_{{\gamma }_1\cdots {\gamma }_{\beta }}\left(t_{\epsilon }\right)={\pi }_{\beta \epsilon }}$$

$${\displaystyle \psi \left(z_{\pi }\left(t\right)\right)\le 0,\tau \left(z_{\pi }\left(t\right)\right)=0,t\in \Omega ,\beta =\overline{1,\mu -1},}$$

with ${\displaystyle \epsilon \in \{0,1\}}$.

In this paper, we are looking for necessary efficiency conditions for the foregoing multiobjective variational problem (MVP). The next section introduces the necessary mathematical elements which will be used to state the main results.

2. Preliminaries

Let us start with the case of one functional, considering the next scalar problem (SP),

$$\begin{gathered} {\displaystyle \underset{\pi (·)}{min}\left\{\Theta \left(\pi (·)\right)={\int }_{\Omega }Y\left(z_{\pi }\left(t\right)\right)dv\right\} \\ \mathrm{subject}\mathrm{to}\pi \left(·\right)\in \text{Q}\left(\Omega \right).} \end{gathered}$$

Consider the auxiliary Lagrange functional ${\displaystyle B}$, defined as

$${\displaystyle B(z_{\pi }\left(t\right),e\left(t\right),u\left(t\right),h):=hY\left(z_{\pi }\left(t\right)\right)}$$

$${\displaystyle +e^a\left(t\right){\psi }_a\left(z_{\pi }\left(t\right)\right)+u^{\zeta }\left(t\right){\tau }_{\zeta }\left(z_{\pi }\left(t\right)\right),}$$

(with summation over the repeated indices) that allows us to establish necessary conditions of efficiency for (SP). We must note that the next theorem includes functional perturbation, first variation derivation, and integration by parts associated with higher-order derivatives.

Theorem 1.

Consider that ${\displaystyle {\pi }^0}$ is an optimal solution and ${\displaystyle Y,\psi ,\tau }$ are ${\displaystyle C^{\mu +1}}$-class functions. Then, there exist ${\displaystyle h}$ and the following functions (piecewise smooth) ${\displaystyle e\left(t\right)}$ and ${\displaystyle u\left(t\right)}$, fulfilling

$${\displaystyle \frac{\partial B}{\partial \pi }(z_{{\pi }^0}\left(t\right),e\left(t\right),u\left(t\right),h)}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\frac{\partial B}{\partial {\pi }_{{\gamma }_1}}(z_{{\pi }^0}\left(t\right),e\left(t\right),u\left(t\right),h)}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\frac{\partial B}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}(z_{{\pi }^0}\left(t\right),e\left(t\right),u\left(t\right),h)=0}$$

$$(\mathit{higher-order} \mathit{Euler-Lagrange} \mathit{PDEs})$$

$${\displaystyle e\left(t\right)\psi \left(z_{{\pi }^0}\left(t\right)\right)=0,e\left(t\right)\ge 0,(\forall )t\in \Omega .}$$

The previous result represents a generalization of Valentine’s necessary conditions (see [22]).

Definition 1.

The optimal solution ${\displaystyle {\pi }^0(·)}$ of problem (SP) is called normal optimal solution if ${\displaystyle h\ne 0}$.

Further, we can assume that ${\displaystyle h=1}$.

In this work, we are interested in finding the necessary conditions of efficiency of the considered problem (MVP) in ${\displaystyle \text{Q}\left(\Omega \right)}$. In this respect, we give the following definition.

Definition 2.

A feasible solution ${\displaystyle {\pi }^0(·)\in \text{Q}\left(\Omega \right)}$ is named efficient solution in (MVP) if there exists no other ${\displaystyle \pi (·)\in \text{Q}\left(\Omega \right)}$ satisfying ${\displaystyle Q\left(\pi (·)\right)⪯Q\left({\pi }^0(·)\right)}$.

3. Main Results

To develop our theory, we establish the following results. The next lemma is a connection of the weighted-sum scalarization and the Geoffrion efficiency concept (see Definition 2).

Lemma 1.

The feasible solution ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ is an efficient solution of the problem (MVP) if and only if ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ is an optimal solution of every ${\displaystyle A_{\vartheta }\left({\pi }^0\right),\vartheta =\overline{1,a}}$,

$${\displaystyle \underset{\pi (·)}{min}{\int }_{\Omega }{\sigma }_{\vartheta }\left(z_{\pi }\left(t\right)\right)dv}$$

$$\mathit{subject}\mathit{to}$$

$${\displaystyle \pi \left(·\right)\in \text{Q}\left(\Omega \right),{\int }_{\Omega }{\sigma }_{\nu }\left(z_{\pi }\left(t\right)\right)dv\le {\int }_{\Omega }{\sigma }_{\nu }\left(z_{{\pi }^0}\left(t\right)\right)dv}$$

for ${\displaystyle \nu =\overline{1,a},\nu \ne \vartheta }$.

Proof. 

“${\displaystyle ⟹}$” Consider ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ an efficient solution of (MVP). Let us suppose there exists ${\displaystyle \mu \in \{1,\dots ,a\}}$ such that ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ is not an optimal solution of ${\displaystyle A_{\mu }\left({\pi }^0\right)}$. Thus, there exists ${\displaystyle y\left(·\right)\in \text{Q}\left(\Omega \right)}$ fulfilling

$${\displaystyle {\int }_{\Omega }{\sigma }_{\nu }\left(z_y\left(t\right)\right)dv\le {\int }_{\Omega }{\sigma }_{\nu }\left(z_{{\pi }^0}\left(t\right)\right)dv}$$

for ${\displaystyle \nu =\overline{1,a},\nu \ne \mu }$, and

$${\displaystyle {\int }_{\Omega }{\sigma }_{\mu }\left(z_y\left(t\right)\right)dv<{\int }_{\Omega }{\sigma }_{\mu }\left(z_{{\pi }^0}\left(t\right)\right)dv.}$$

This contradicts the efficiency of the function ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ in (MVP). Consequently, we proved the direct implication.

“${\displaystyle ⟸}$” Let ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ be an optimal solution of ${\displaystyle A_{\vartheta }\left({\pi }^0\right),\vartheta =\overline{1,a}}$. Assume that ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ is not an efficient solution of (MVP). Thus, there exists ${\displaystyle y\left(·\right)\in \text{Q}\left(\Omega \right)}$ fulfilling

$${\displaystyle {\int }_{\Omega }{\sigma }_{\nu }\left(z_y\left(t\right)\right)dv\le {\int }_{\Omega }{\sigma }_{\nu }\left(z_{{\pi }^0}\left(t\right)\right)dv,\nu =\overline{1,a}}$$

and there exists ${\displaystyle \mu \in \{1,\dots ,a\}}$ such that

$${\displaystyle {\int }_{\Omega }{\sigma }_{\mu }\left(z_y\left(t\right)\right)dv<{\int }_{\Omega }{\sigma }_{\mu }\left(z_{{\pi }^0}\left(t\right)\right)dv.}$$

But, ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ minimizes ${\displaystyle {\int }_{\Omega }{\sigma }_{\mu }\left(z_{\pi }\left(t\right)\right)dv}$ on the feasible solution set of problem ${\displaystyle A_{\mu }\left({\pi }^0\right)}$. The proof is complete.   □

The next lemma, assuming the necessary constraint qualifications, takes into account the Lagrange multipliers. Similar arguments are considered for Pontryagin’s Principle in PDE control systems (see Udrişte and Matei [20]).

Lemma 2.

Consider ${\displaystyle \vartheta \in \{1,\dots ,a\}}$ fixed and ${\displaystyle {\pi }^0\left(·\right)\in \text{Q}\left(\Omega \right)}$ an optimal solution for ${\displaystyle A_{\vartheta }\left({\pi }^0\right)}$. Then there exist ${\displaystyle h_{\nu \vartheta }\ge 0}$ and ${\displaystyle e_{\vartheta }\left(t\right)}$, ${\displaystyle u_{\vartheta }\left(t\right)}$ (piecewise smooth functions) such that

$${\displaystyle \sum _{\nu =1}^a{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\left(t\right)\right)+e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\left(t\right)\right)}$$

$${\displaystyle +u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\left(t\right)\right)-\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1}^a{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\left(t\right)\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1}^a{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}=0}$$

$${\displaystyle \left(higherorderEuler\text{-}LagrangePDEs\right)}$$

$${\displaystyle e_{\vartheta }\left(t\right)\psi \left(z_{{\pi }^0}\left(t\right)\right)=0,e_{\vartheta }\left(t\right)\ge 0,(\forall )t\in \Omega .}$$

Proof. 

Consider ${\displaystyle M_{\vartheta }^0:={\int }_{\Omega }{\sigma }_{\vartheta }\left(z_{{\pi }^0}\left(t\right)\right)dv=\underset{\pi (·)}{min}{\int }_{\Omega }{\sigma }_{\vartheta }\left(z_{\pi }\left(t\right)\right)dv}$, ${\displaystyle \vartheta =\overline{1,a}}$. Define the ${\displaystyle C^{\mu +1}}$-class functions, ${\displaystyle {\Psi }_{\nu }:\Omega \times {\mathbb{R}}^{n(\mu +1)}\to \mathbb{R},{\Psi }_{\nu }\left(z_{\pi }\left(t\right)\right)\ge 0,\nu =\overline{1,a},\nu \ne \vartheta }$, as follows

$${\displaystyle F_{\nu }\left(\pi (·)\right):={\int }_{\Omega }\left[{\sigma }_{\nu }\left(z_{\pi }\left(t\right)\right)-M_{\nu }^0+{\Psi }_{\nu }\left(z_{\pi }\left(t\right)\right)\right]dv}$$

$${\displaystyle =0.}$$

Thus, the problem ${\displaystyle A_{\vartheta }\left({\pi }^0\right)}$, ${\displaystyle \vartheta \in \{1,\dots ,a\}}$ fixed, becomes

$${\displaystyle \underset{\pi (·)}{max}{\int }_{\Omega }{\sigma }_{\vartheta }\left(z_{\pi }\right)dv}$$

$$\mathrm{subject}\mathrm{to}$$

$${\displaystyle \pi \in \text{Q}\left(\Omega \right),F_{\nu }\left(\pi (·)\right)=0}$$

$${\displaystyle {\Psi }_{\nu }\left(z_{\pi }\right)\ge 0,\nu =\overline{1,a},\nu \ne \vartheta }$$

or

$${\displaystyle \underset{\pi (·)}{max}\{{\int }_{\Omega }{\sigma }_{\vartheta }\left(z_{\pi }\right)}$$

$${\displaystyle +\sum _{\nu =1;\nu \ne \vartheta }^a{h}_{\nu \vartheta }\left[{\sigma }_{\nu }\left(z_{\pi }\right)-M_{\nu }^0+{\Psi }_{\nu }\left(z_{\pi }\right)\right]dv\}}$$

$$\mathrm{subject}\mathrm{to}$$

$${\displaystyle \pi \in \text{Q}\left(\Omega \right),{\Psi }_{\nu }\left(z_{\pi }\right)\ge 0,\nu =\overline{1,a},\nu \ne \vartheta .}$$

or, equivalently,

$${\displaystyle \underset{\pi (·)}{max}\{{\int }_{\Omega }{\sigma }_{\vartheta }\left(z_{\pi }\right)}$$

(1)

$${\displaystyle +\sum _{\nu =1;\nu \ne \vartheta }^a{h}_{\nu \vartheta }\left[{\sigma }_{\nu }\left(z_{\pi }\right)-M_{\nu }^0+{\Psi }_{\nu }\left(z_{\pi }\right)\right]dv\}}$$

$$\mathrm{subject}\mathrm{to}$$

$${\displaystyle \pi \left(·\right)\in \text{Q}\left(\Omega \right),-{\Psi }_{\nu }\left(z_{\pi }\right)\le 0,\nu =\overline{1,a},\nu \ne \vartheta .}$$

Let consider the following Lagrangian,

$${\displaystyle R_{\vartheta }(z_{\pi },e_{\vartheta },u_{\vartheta },{\gamma }_{\vartheta },a_{\nu }):={\gamma }_{\vartheta }{\sigma }_{\vartheta }\left(z_{\pi }\right)}$$

$${\displaystyle +{\gamma }_{\vartheta }\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{h}_{\nu \vartheta }\left[{\sigma }_{\nu }\left(z_{\pi }\right)-M_{\nu }^0+{\Psi }_{\nu }\left(z_{\pi }\right)\right]\right\}}$$

$${\displaystyle +e_{\vartheta }\left(t\right)\psi \left(z_{\pi }\right)+u_{\vartheta }\left(t\right)\tau \left(z_{\pi }\right)-\sum _{\nu =1;\nu \ne \vartheta }^a{a}_{\nu }\left(t\right){\Psi }_{\nu }\left(z_{\pi }\right),}$$

where ${\displaystyle {\gamma }_{\vartheta }\in \mathbb{R},{\gamma }_{\vartheta }\ge 0}$, and ${\displaystyle e_{\vartheta }:\Omega \to {\mathbb{R}}^k,u_{\vartheta }:\Omega \to {\mathbb{R}}^r}$, ${\displaystyle a_{\nu }:\Omega \to \mathbb{R},a_{\nu }\ge 0,\nu =\overline{1,a},\nu \ne \vartheta }$ are piecewise smooth functions. Since ${\displaystyle {\pi }^0}$ is an optimal solution in ${\displaystyle \left(1\right)}$, the following Valentine’s necessary conditions hold

$${\displaystyle \frac{\partial R_{\vartheta }}{\partial \pi }(z_{{\pi }^0},e_{\vartheta },u_{\vartheta },{\gamma }_{\vartheta },a_{\nu })-\frac{\partial }{\partial t^{{\gamma }_1}}\frac{\partial R_{\vartheta }}{\partial {\pi }_{{\gamma }_1}}(z_{{\pi }^0},e_{\vartheta },u_{\vartheta },{\gamma }_{\vartheta },a_{\nu })}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\frac{\partial R_{\vartheta }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}(z_{{\pi }^0},e_{\vartheta },u_{\vartheta },{\gamma }_{\vartheta },a_{\nu })=0}$$

$${\displaystyle e_{\vartheta }\left(t\right)\psi \left(z_{{\pi }^0}\left(t\right)\right)=0,e_{\vartheta }\left(t\right)\ge 0,t\in \Omega }$$

$${\displaystyle a_{\nu }\left(t\right){\Psi }_{\nu }\left(z_{{\pi }^0}\left(t\right)\right)=0,a_{\nu }\left(t\right)\ge 0,\nu =\overline{1,a},\nu \ne \vartheta }$$

$${\displaystyle {\gamma }_{\vartheta }\ge 0,h_{\nu \vartheta }\ge 0,\nu =\overline{1,a},\nu \ne \vartheta .}$$

Concretely, we have

$${\displaystyle {\gamma }_{\vartheta }\frac{\partial {\sigma }_{\vartheta }}{\partial \pi }\left(z_{{\pi }^0}\right)+\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\left[\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)+\frac{\partial {\Psi }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)\right]}$$

$${\displaystyle +e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\right)}$$

$${\displaystyle -\sum _{\nu =1;\nu \ne \vartheta }^a{a}_{\nu }\left(t\right)\frac{\partial {\Psi }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)-\frac{\partial }{\partial t^{{\gamma }_1}}\left\{{\gamma }_{\vartheta }\frac{\partial {\sigma }_{\vartheta }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\left[\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+\frac{\partial {\Psi }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right]\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{a}_{\nu }\left(t\right)\frac{\partial {\Psi }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{{\gamma }_{\vartheta }\frac{\partial {\sigma }_{\vartheta }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\frac{\partial {\Psi }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$$+{(-1)}^{\mu +1}{\displaystyle \frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{a}_{\nu }\left(t\right){\displaystyle \frac{\partial {\Psi }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}}\left(z_{{\pi }^0}\right)\right\}=0,$$

or, equivalently,

$${\displaystyle {\gamma }_{\vartheta }\frac{\partial {\sigma }_{\vartheta }}{\partial \pi }\left(z_{{\pi }^0}\right)+\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)}$$

(2)

$${\displaystyle +\sum _{\nu =1;\nu \ne \vartheta }^a\left[{\gamma }_{\vartheta }{h}_{\nu \vartheta }-a_{\nu }\left(t\right)\right]\frac{\partial {\Psi }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)}$$

$${\displaystyle +e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\right)}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{{\gamma }_{\vartheta }\frac{\partial {\sigma }_{\vartheta }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a\left[{\gamma }_{\vartheta }{h}_{\nu \vartheta }-a_{\nu }\left(t\right)\right]\frac{\partial {\Psi }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{{\gamma }_{\vartheta }\frac{\partial {\sigma }_{\vartheta }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{\gamma }_{\vartheta }{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a\left[{\gamma }_{\vartheta }{h}_{\nu \vartheta }-a_{\nu }\left(t\right)\right]\frac{\partial {\Psi }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle =0.}$$

Next, we write the conditions: ${\displaystyle {\gamma }_{\vartheta }{h}_{\nu \vartheta }-a_{\nu }\left(t\right)=0,\nu =\overline{1,a},\nu \ne \vartheta }$, for any ${\displaystyle t\in \Omega }$, ${\displaystyle {\gamma }_{\vartheta }=h_{ll}\ge 0,h_{\nu \vartheta }={\gamma }_{\vartheta }{h}_{\nu \vartheta }\ge 0,\nu =\overline{1,a},\nu \ne \vartheta }$. By considering ${\displaystyle \left(2\right)}$, it follows

$${\displaystyle h_{ll}\frac{\partial {\sigma }_{\vartheta }}{\partial \pi }\left(z_{{\pi }^0}\right)+\sum _{\nu =1;\nu \ne \vartheta }^a{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)+e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\right)}$$

$${\displaystyle +u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\right)-\frac{\partial }{\partial t^{{\gamma }_1}}\left\{h_{ll}\frac{\partial {\sigma }_{\vartheta }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{h_{ll}\frac{\partial {\sigma }_{\vartheta }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1;\nu \ne \vartheta }^a{h}_{\nu \vartheta }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{e_{\vartheta }\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)+u_{\vartheta }\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle =0}$$

and the proof is complete.   □

Definition 3.

The point ${\displaystyle {\pi }^0(·)\in \text{Q}\left(\Omega \right)}$ is called a normal efficient solution in (MVP) if it is a normal optimal solution for at least one ${\displaystyle A_{\vartheta }\left({\pi }^0\right),\vartheta =\overline{1,a}}$.

Now, we establish the main result of this study, namely, the normal necessary efficiency conditions of the program (MVP). The presence of higher-order derivatives introduces additional boundary terms and complexity. This result can be reduced to the classical form, as well.

Theorem 2

([Normal] necessary efficiency conditions for (MVP)). If ${\displaystyle {\pi }^0(·)\in \text{Q}\left(\Omega \right)}$ is a [normal] efficient solution in (MVP), then there are ${\displaystyle h\in {\mathbb{R}}^a}$, ${\displaystyle e:\Omega \to {\mathbb{R}}^k}$ and ${\displaystyle u:\Omega \to {\mathbb{R}}^r}$ fulfilling the relations:

$${\displaystyle \sum _{\nu =1}^a{h}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\left(t\right)\right)+e\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\left(t\right)\right)}$$

$${\displaystyle +u\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\left(t\right)\right)-\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1}^a{h}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{e\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\left(t\right)\right)+u\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1}^a{h}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{e\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{u\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\left(t\right)\right)\right\}=0}$$

$${\displaystyle \left(higherorderEuler\text{-}LagrangePDEs\right)}$$

$${\displaystyle e\left(t\right)\psi \left(z_{{\pi }^0}\left(t\right)\right)=0,e\left(t\right)\ge 0,(\forall )t\in \Omega }$$

$${\displaystyle h\ge 0,q^{t}h=1,q^t=(1,1,\dots ,1)\in {\mathbb{R}}^a.}$$

Proof. 

By using Lemma 1, we get ${\displaystyle {\pi }^0(·)\in \text{Q}\left(\Omega \right)}$ is an optimal solution in ${\displaystyle A_{\vartheta }\left({\pi }^0\right),\vartheta =\overline{1,a}}$. Thus, if ${\displaystyle {\pi }^0(·)\in \text{Q}\left(\Omega \right)}$ is [normal] optimal solution of ${\displaystyle A_{\vartheta }\left({\pi }^0\right),\vartheta \in \{1,\dots ,a\}}$ fixed, then Lemma 2 holds [${\displaystyle h_{ll}=1}$]. By summation over ${\displaystyle \vartheta =\overline{1,a}}$ of all condition given in Lemma 2 and by setting

$${\displaystyle \sum _{\vartheta =1}^a{h}_{\nu \vartheta }={\tilde{h}}_{\nu },\sum _{\vartheta =1}^a{e}_{\vartheta }\left(t\right)=\tilde{a}\left(t\right),\sum _{\vartheta =1}^a{u}_{\vartheta }\left(t\right)=\tilde{u}\left(t\right),}$$

we obtain

$${\displaystyle \sum _{\nu =1}^a{\tilde{h}}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)+\tilde{a}\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\right)+\tilde{u}\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\right)}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1}^a{\tilde{h}}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+\tilde{a}\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\tilde{u}\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\dots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1}^a{\tilde{h}}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\dots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\dots \partial t^{{\gamma }_{\mu }}}\left\{\tilde{a}\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\dots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)+\tilde{u}\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\dots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}=0}$$

$${\displaystyle \tilde{a}\left(t\right)\psi \left(z_{{\pi }^0}\left(t\right)\right)=0,\tilde{a}\left(t\right)\ge 0,(\forall )t\in \Omega }$$

$${\displaystyle {\tilde{h}}_{\nu }\ge 0,[{\tilde{h}}_{\nu }\ge 1].}$$

By dividing with ${\displaystyle D=\sum _{\nu =1}^a{\tilde{h}}_{\nu }\ge 1}$ and denoting ${\displaystyle h_{\nu }={\tilde{h}}_{\nu }/D,e\left(t\right)=\tilde{a}\left(t\right)/D}$, ${\displaystyle u\left(t\right)=\tilde{u}\left(t\right)/D}$, we obtain

$${\displaystyle \sum _{\nu =1}^a{h}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial \pi }\left(z_{{\pi }^0}\right)+e\left(t\right)\frac{\partial \psi }{\partial \pi }\left(z_{{\pi }^0}\right)+u\left(t\right)\frac{\partial \tau }{\partial \pi }\left(z_{{\pi }^0}\right)}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{\sum _{\nu =1}^a{h}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)+e\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle -\frac{\partial }{\partial t^{{\gamma }_1}}\left\{u\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +\dots +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{\sum _{\nu =1}^a{h}_{\nu }\frac{\partial {\sigma }_{\nu }}{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{e\left(t\right)\frac{\partial \psi }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}}$$

$${\displaystyle +{(-1)}^{\mu }\frac{{\partial }^{\mu }}{\partial t^{{\gamma }_1}\cdots \partial t^{{\gamma }_{\mu }}}\left\{u\left(t\right)\frac{\partial \tau }{\partial {\pi }_{{\gamma }_1\cdots {\gamma }_{\mu }}}\left(z_{{\pi }^0}\right)\right\}=0}$$

$${\displaystyle e\left(t\right)\psi \left(z_{{\pi }^0}\left(t\right)\right)=0,e\left(t\right)\ge 0,(\forall )t\in \Omega }$$

$${\displaystyle h\ge 0,q^{t}h=1,q^t=(1,1,\dots ,1)\in {\mathbb{R}}^a}$$

and the proof is complete.   □

Application.  Find the extremals of the following functional

$${\displaystyle I\left(\pi (·)\right)=\frac{1}{2}{\int }_{\Omega }\left({\pi }_x^2+{\pi }_y^2\right)dxdy,}$$

subject to ${\displaystyle 2{\pi }_x+3{\pi }_y+8\pi =0}$ and the boundary conditions ${\displaystyle \pi (0,0)=0,\pi (1,1)=1}$, where ${\displaystyle \Omega ={[0,1]}^2}$.

Solution. Consider the following Lagrangian

$${\displaystyle B=\frac{1}{2}\left({\pi }_x^2+{\pi }_y^2\right)+u(x,y)\left(2{\pi }_x+3{\pi }_y+8\pi \right).}$$

Since

$${\displaystyle \frac{\partial B}{\partial \pi }=8u,\frac{\partial B}{\partial {\pi }_x}={\pi }_x+2u,\frac{\partial B}{\partial {\pi }_y}={\pi }_y+3u,}$$

the extremals are described by the following system of PDEs

$${\displaystyle 8u-\frac{\partial }{\partial x}\left({\pi }_x+2u\right)-\frac{\partial }{\partial y}\left({\pi }_y+3u\right)=0,2{\pi }_x+3{\pi }_y+8\pi =0.}$$

The first-order PDE ${\displaystyle 2{\pi }_x+3{\pi }_y+8\pi =0}$ admits the general solution ${\displaystyle \pi (x,y)=e^{-4x}f(3x-2y)}$, where f is an arbitrary $C^2$-class function. Replacing in the second-order PDE, we find a first-order PDE with $u(x,y)$ as unknown.

Remark 1.

The higher-order Euler–Lagrange conditions have, of course, an impact on solution smoothness. Moreover, computational challenges (e.g., solving high-order PDE systems) inevitably appear (see, for instance, Treanţă [5]).

4. Conclusions

In this paper, the authors investigated a new family of optimization models involving higher order partial derivatives. Concretely, necessary efficiency criteria have been stated and proved for this class of variational problems. In this regard, a specific and complex framework has been adopted. As further developments associated with the current paper, the author would recommend the extension of these results for curvilinear type functionals which are path-independent. Another new research direction is the study of saddle-point criteria for this class of extremization models.