Optimality for Control Problem with PDEs of Second-Order as Constraints

Abstract

This paper deals with a class of second-order partial differential equation (in short, PDE) constrained optimal control problems. More specifically, by using appropriate variational techniques, we state necessary conditions of optimality associated with this class of optimization problems, defined by controlled curvilinear integral cost functionals involving partial derivatives of second-order. The importance of the considered problem is provided by its applications in mechanics and physics. Compared with other research works, here we develop a new mathematics context that extends the results obtained so far, both through the use of controlled curvilinear integrals and also by considering partial derivatives of second-order. In addition, to emphasize the usefulness of the main results, an illustrative example is provided.

1. Introduction

Since Optimal Control Theory and Calculus of Variations are two mathematical fields with strong and important connections, many researchers have studied these areas and the connections between them, obtaining remarkable results (see, for instance, the works of Friedman [1], Hestenes [2], Kendall [3], Udrişte [4], Petrat and Tumulka [5], Treanţă [6] and Deckert and Nickel [7]). The previously mentioned research papers dealt with problems in several time variables. In the last period, the study of multi-dimensional (that is, in several time variables) optimization problems (with many important applications in various branches of mathematical sciences, decision problems in management science, web access problems, engineering design, portfolio selection, game theory, query optimization in databases, and so on), has been conducted by Mititelu and Treanţă [8], Treanţă [6,9,10,11,12]. More specifically, they have studied several optimization problems formulated by curvilinear and multiple integral functionals with mixed and isoperimetric constraints with first-order m-flow type PDEs and PDIs. Quite recently, Treanţă [10] has focused the study on the optimization problems with various cost functionals (defined by second-order Lagrangians) subject to varied constraints. For other ideas on this topic, the reader can consult the works of Arisawa and Ishii [13], Lai and Motta [14], Shi et al. [15], An et al. [16], Zhao et al. [17], Hung et al. [18], Tajadodi [19], Xiaobing et al. [20], Chen et al. [21], Lü and Chen [22], Yin et al. [23].

In the present paper, we introduce a new type of PDE constrained optimal control problem, formulated by curvilinear integrals which involve partial derivatives of second-order. For instance, Udrişte and Ţevy [4] established the Euler–Lagrange PDEs associated with multi-variate non-controlled Lagrangians of first-order defined by curvilinear or multiple integrals. Here, we develop a new math context that extends the ones given in Hestenes [2], Treanţă [10], or Udrişte and Ţevy [4] both through the use of controlled curvilinear integrals but also by the considering of partial derivatives of second-order. A similar study, using the Lagrange-type variational approach and partial derivatives of second-order, was recently established by Treanţă [11]. The main difference is the presence of multiple (not curvilinear) integrals as cost functionals. Due to the considered Lagrange-type variational approach, the closeness (complete integrability) conditions associated with the considered Lagrange 1-forms generate new PDEs which can be considered as new constraints for the problem under study. In addition, in this paper, the method for formulating the proof for the main result derived represents another novelty element. Thus, this paper represents an important work for researchers in the field of engineering and applied sciences (the functionals of curvilinear integral-type compute the mechanical work performed by a variable force), where second-order PDEs and curvilinear integrals are involved.

Further, this paper is organized in the following way. Section 2 formulates the optimal control problem. In addition, we state the principal result associated with this article (see Theorem 1). This theorem provides the necessary optimality conditions of the considered second-order PDE constrained control problem. In addition, in the final part of this section, we present an illustrative application. Section 3 concludes the present paper.

2. Control Problem with PDEs of Second-Order as Constraints

We start with ${\displaystyle {\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right),\pi =\overline{1,m}}$, as functions of ${\displaystyle C^3}$-class, which we call multi-variate controlled Lagrangians of second-order, with ${\displaystyle t=\left(t^{\delta }\right)=(t^1,\cdots ,t^m)}$$\in \Delta \subset {\mathbb{R}}_{+}^m,b=\left(b^i\right)=\left(b^1,\cdots ,b^n\right):\Delta \to {\mathbb{R}}^n$ is a function of $C^4$-class (named state variable), and $v=\left(v^{\vartheta }\right)=\left(v^1,\cdots ,v^k\right):\Delta \to {\mathbb{R}}^k$ represents a continuous function (or piecewise continuous), named the control variable. In addition, we use the notations ${\displaystyle b_{\delta }\left(t\right):=\frac{\partial b}{\partial t^{\delta }}\left(t\right),b_{\delta \varrho }\left(t\right):=\frac{{\partial }^{2}b}{\partial t^{\delta }\partial t^{\varrho }}\left(t\right),\delta ,\varrho \in \{1,\dots ,m\}}$, and notice that ${\displaystyle \Delta =[t_0,t_1]}$ (multi-variate interval included in ${\displaystyle {\mathbb{R}}_{+}^m}$). Moreover, we assume that the previous Lagrangians provide a closed controlled Lagrange 1-form

$${\displaystyle {\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)d{t}^{\pi }}$$

(with summation on repeated indices), which gives the next controlled curvilinear integral (which is independent of the path)

$${\displaystyle K\left(b(·),v(·)\right)={\int }_C{\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)d{t}^{\pi },}$$

(1)

where ${\displaystyle C}$ is a curve that is included in ${\displaystyle \Delta }$ and links the points ${\displaystyle t_0,t_1\in {\mathbb{R}}_{+}^m}$.

Control problem with PDEs of second-order as constraints.Find ${\displaystyle (b^{*},v^{*})}$ such that to minimize the functional $\left(1\right)$, for all the functions ${\displaystyle (b,v)}$ which satisfy

$${\displaystyle b\left(t_0\right)=b_0,b\left(t_1\right)=b_1,b_{\kappa }\left(t_0\right)={\tilde{b}}_{\kappa 0},b_{\kappa }\left(t_1\right)={\tilde{b}}_{\kappa 1}}$$

and the controlled second-order PDE constraints (partial speed–acceleration constraints) defined as follows:

$${\displaystyle F_{\pi }^{\iota }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)=0,\iota =1,2,\cdots ,r\le n,\pi =1,2,\cdots ,m.}$$

To study the above control problem ${\displaystyle \left(1\right)}$, together with the controlled second-order PDE constraints, we introduce ${\displaystyle \omega =\left({\omega }_{\iota }\left(t\right)\right)}$ and consider new Lagrangians (in the following, see Einstein summation)

$${\displaystyle {\mathsf{\Theta }}_{1\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),\omega \left(t\right),t\right)={\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)}$$

$${\displaystyle +{\omega }_{\iota }\left(t\right)F_{\pi }^{\iota }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right),\pi =\overline{1,m},}$$

modifying the initial control problem (with second-order PDE constraints) into a partial speed–acceleration unconstrained control problem

$${\displaystyle \underset{\left(b\right(·),v(·),\omega (·\left)\right)}{min}{\int }_C{\mathsf{\Theta }}_{1\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),\omega \left(t\right),t\right)d{t}^{\pi }}$$

(2)

$${\displaystyle b\left(t_q\right)=b_q,b_{\kappa }\left(t_q\right)={\tilde{b}}_{\kappa q},q=0,1,}$$

if the Lagrange 1-form ${\mathsf{\Theta }}_{1\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),\omega \left(t\right),t\right)d{t}^{\pi }$ is complete integrable.

By using Lagrange theory, an extreme of ${\displaystyle \left(1\right)}$ can be found among the extremes of ${\displaystyle \left(2\right)}$.

For foumulating the necessary conditions of optimality for our control problem, we will use the Saunders’s multi-index notation (see Saunders [24], Treanţă [11,12]). In short, a multi-index represents an ${\displaystyle m}$-tuple ${\displaystyle U}$ consisting of natural numbers. We denote ${\displaystyle U\left(\delta \right)}$ as the components of ${\displaystyle U}$, where ${\displaystyle \delta }$ represents an ordinary index, with ${\displaystyle 1\le \delta \le m}$. We define ${\displaystyle 1_{\delta }}$ as ${\displaystyle 1_{\delta }\left(\delta \right)=1,1_{\delta }\left(\varrho \right)=0}$ for ${\displaystyle \delta \ne \varrho }$, and ${\displaystyle (U\pm V)\left(\delta \right)=U\left(\delta \right)\pm V\left(\delta \right)}$ (although the substraction might not be a multi-index). Define the length as ${\displaystyle \mid U\mid =\sum _{\delta =1}^{m}U\left(\delta \right)}$, and the factorial by ${\displaystyle U!=\prod _{\delta =1}^m\left(U\left(\delta \right)\right)!}$. In addition, the number of different indices is given by ${\displaystyle \{{\delta }_1,{\delta }_2,\dots ,{\delta }_k\},{\delta }_j\in \{1,2,\dots ,m\}}$, ${\displaystyle j=\overline{1,k}}$, is

$${\displaystyle \tau ({\delta }_1,{\delta }_2,\dots ,{\delta }_k)=\frac{\mid 1_{{\delta }_1}+1_{{\delta }_2}+\dots +1_{{\delta }_k}\mid !}{(1_{{\delta }_1}+1_{{\delta }_2}+\dots +1_{{\delta }_k})!}.}$$

In the next theorem, we present optimality conditions which are necessary (not sufficient) for a point to be an extreme of the studied control problem with PDEs of second-order as constraints.

Theorem 1.

Let ${\displaystyle \left(b^{*}(·),v^{*}(·),{\omega }^{*}(·)\right)}$ be a solution for the considered control problem with PDEs of second-order as constraints (see ${\displaystyle \left(2\right)}$). Then

$${\displaystyle \left(b^{*}(·),v^{*}(·),{\omega }^{*}(·)\right)}$$

solves the next PDEs of Euler–Lagrange type

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^i}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^i}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^i}=0,i=\overline{1,n},\pi =\overline{1,m}}$$

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\kappa }^{\vartheta }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\delta \varrho }^{\vartheta }}=0,\vartheta =\overline{1,k},\pi =\overline{1,m}}$$

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota ,\kappa }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota ,\delta \varrho }}=0,\iota =\overline{1,r},\pi =\overline{1,m},}$$

where ${\displaystyle {\omega }_{\iota ,\kappa }:=\frac{\partial {\omega }_{\iota }}{\partial t^{\kappa }},{\omega }_{\iota ,\delta \varrho }:=\frac{{\partial }^2{\omega }_{\iota }}{\partial t^{\delta }\partial t^{\varrho }},v_{\delta \varrho }^{\vartheta }:=\frac{{\partial }^2{v}^{\vartheta }}{\partial t^{\delta }\partial t^{\varrho }},\delta ,\varrho ,\kappa \in \{1,2,\dots ,m\}}$.

Proof. 

Let us consider that ${\displaystyle \left(b\left(t\right),v\left(t\right),\omega \left(t\right)\right)}$ is a solution for the considered control problem with PDEs of second-order as constraints (see ${\displaystyle \left(2\right)}$). Now, we define ${\displaystyle b\left(t\right)+\sigma \chi \left(t\right)}$ to be a variation for ${\displaystyle b\left(t\right)}$, with ${\displaystyle \chi \left(t_0\right)=\chi \left(t_1\right)=0,{\chi }_{\eta }\left(t_0\right)={\chi }_{\eta }\left(t_1\right)=0,\eta \in \{1,2,\dots ,m\}}$ (see ${\displaystyle {\chi }_{\eta }:=\frac{\partial \chi }{\partial t^{\eta }}}$),${\displaystyle \omega \left(t\right)+\sigma \mathsf{f}\left(t\right)}$ to be a variation for ${\displaystyle \omega \left(t\right)}$, with ${\displaystyle \mathsf{f}\left(t_0\right)=\mathsf{f}\left(t_1\right)=0}$, and ${\displaystyle v\left(t\right)+\sigma \mathsf{m}\left(t\right)}$ to be a variation for ${\displaystyle v\left(t\right)}$, satisfying ${\displaystyle \mathsf{m}\left(t_0\right)=\mathsf{m}\left(t_1\right)=0}$. The “small” variations ${\displaystyle \chi ,\mathsf{f},\mathsf{m}}$, and the “small” parameter ${\displaystyle \sigma }$ are used in the next variational techniques. By using the above variations, the objective functional becomes an integral depending on a parameter

$${\displaystyle P\left(\sigma \right)={\int }_C{\mathsf{\Theta }}_{1\pi }(b\left(t\right)+\sigma \chi \left(t\right),b_{\kappa }\left(t\right)+\sigma {\chi }_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right)+\sigma {\chi }_{\delta \varrho }\left(t\right),v\left(t\right)+\sigma \mathsf{m}\left(t\right),}$$

$${\displaystyle \omega \left(t\right)+\sigma \mathsf{f}\left(t\right),t)d{t}^{\pi }.}$$

The following relation holds (according to our hypothesis)

$${\displaystyle 0=\frac{d}{d\sigma }P\left(\sigma \right){|}_{\sigma =0}={\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^j}{\chi }^j+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}{\chi }_{\kappa }^j+\frac{1}{\tau (\delta ,\varrho )}\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}{\chi }_{\delta \varrho }^j+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}{\mathsf{m}}^{\vartheta }}$$

$$+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota }}{\mathsf{f}}^{\iota })d{t}^{\pi }$$

$${\displaystyle =BT+{\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^j}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}){\chi }^{j}d{t}^{\pi }}$$

$${\displaystyle +{\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\kappa }^{\vartheta }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\delta \varrho }^{\vartheta }}){\mathsf{m}}^{\vartheta }d{t}^{\pi }}$$

$${\displaystyle +{\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota ,\kappa }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota ,\delta \varrho }}){\mathsf{f}}^{\iota }d{t}^{\pi }.}$$

Now, by using the integration by parts, it results that

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}{\chi }_{\kappa }^j=-{\chi }^j{D}_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}+D_{\kappa }\left(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}{\chi }^j\right),}$$

$${\displaystyle \frac{1}{\tau (\delta ,\varrho )}\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}{\chi }_{\delta \varrho }^j=\frac{1}{\tau (\delta ,\varrho )}\left[{\chi }^j{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}-D_{\delta }\left({\chi }^j{D}_{\varrho }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}\right)+D_{\varrho }\left(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}{\chi }_{\delta }^j\right)\right].}$$

We find that the boundary elements ${\displaystyle BT}$ are zero (${\displaystyle \chi \left(t_q\right)=\mathsf{m}\left(t_q\right)=\mathsf{f}\left(t_q\right)=0,{\chi }_{\eta }\left(t_q\right)=0}$, $q=0,1$), if we apply the equalities

$${\displaystyle D_{\kappa }\left(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}{\chi }^j\right)=D_{\pi }\left(\frac{\partial {\mathsf{\Theta }}_{1\kappa }}{\partial b_{\kappa }^j}{\chi }^j\right),}$$

$${\displaystyle D_{\delta }\left({\chi }^j{D}_{\varrho }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}\right)=D_{\pi }\left({\chi }^j{D}_{\varrho }\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b_{\delta \varrho }^j}\right),}$$

$${\displaystyle D_{\varrho }\left(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}{\chi }_{\delta }^j\right)=D_{\pi }\left(\frac{\partial {\mathsf{\Theta }}_{1\varrho }}{\partial b_{\delta \varrho }^j}{\chi }_{\delta }^j\right).}$$

Moreover, we consider that ${\displaystyle \left(b\right(t),v(t),\omega (t\left)\right)}$, solving the considered control problem with PDEs of second-order as constraints (see ${\displaystyle \left(2\right)}$), satisfies the next closeness conditions associated with Lagrange 1-form ${\displaystyle L_{1\pi }}$:

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^i}\frac{\partial b^i}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^i}\frac{\partial b_{\kappa }^i}{\partial t^{\delta }}+\frac{1}{\tau (\delta ,\varrho )}\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^i}\frac{\partial b_{\delta \varrho }^i}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota }}\frac{\partial {\omega }_{\iota }}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}\frac{\partial v^{\vartheta }}{\partial t^{\delta }}}$$

$${\displaystyle =\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b^i}\frac{\partial b^i}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b_{\kappa }^i}\frac{\partial b_{\kappa }^i}{\partial t^{\pi }}+\frac{1}{\tau (\delta ,\varrho )}\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b_{\delta \varrho }^i}\frac{\partial b_{\delta \varrho }^i}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial {\omega }_{\iota }}\frac{\partial {\omega }_{\iota }}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial v^{\vartheta }}\frac{\partial v^{\vartheta }}{\partial t^{\pi }}.}$$

In addition, we accept that ${\displaystyle \chi ,\mathsf{f},\mathsf{m}}$ fulfill the complete inegrability conditions of

$${\displaystyle {\mathsf{\Theta }}_{1\pi }(b\left(t\right)+\sigma \chi \left(t\right),b_{\kappa }\left(t\right)+\sigma {\chi }_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right)+\sigma {\chi }_{\delta \varrho }\left(t\right),v\left(t\right)+\sigma \mathsf{m}\left(t\right),}$$

$${\displaystyle \omega \left(t\right)+\sigma \mathsf{f}\left(t\right),t)d{t}^{\pi }.}$$

This condition implies the next PDEs

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^i}\frac{\partial {\chi }^i}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^i}\frac{\partial {\chi }_{\kappa }^i}{\partial t^{\delta }}+\frac{1}{\tau (\delta ,\varrho )}\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^i}\frac{\partial {\chi }_{\delta \varrho }^i}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota }}\frac{\partial {\mathsf{f}}^{\iota }}{\partial t^{\delta }}+\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}\frac{\partial {\mathsf{m}}^{\vartheta }}{\partial t^{\delta }}}$$

$${\displaystyle =\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b^i}\frac{\partial {\chi }^i}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b_{\kappa }^i}\frac{\partial {\chi }_{\kappa }^i}{\partial t^{\pi }}+\frac{1}{\tau (\delta ,\varrho )}\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial b_{\delta \varrho }^i}\frac{\partial {\chi }_{\delta \varrho }^i}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial {\omega }_{\iota }}\frac{\partial {\mathsf{f}}^{\iota }}{\partial t^{\pi }}+\frac{\partial {\mathsf{\Theta }}_{1\delta }}{\partial v^{\vartheta }}\frac{\partial {\mathsf{m}}^{\vartheta }}{\partial t^{\pi }}.}$$

Finally, we get

$${\displaystyle 0={\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^j}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^j}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^j}){\chi }^{j}d{t}^{\pi }}$$

$${\displaystyle +{\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\kappa }^{\vartheta }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\delta \varrho }^{\vartheta }}){\mathsf{m}}^{\vartheta }d{t}^{\pi }}$$

$${\displaystyle +{\int }_C(\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota ,\kappa }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial {\omega }_{\iota ,\delta \varrho }}){\mathsf{f}}^{\iota }d{t}^{\pi }}$$

and, since the smooth curve ${\displaystyle C}$ is arbitrary, we find the Euler–Lagrange-type PDEs. □

Remark 1.

The PDEs of Euler–Lagrange type, given in Theorem 1, can be formulated in the following form

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b^i}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\kappa }^i}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial b_{\delta \varrho }^i}=0,i=\overline{1,n},\pi =\overline{1,m}}$$

$${\displaystyle \frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v^{\vartheta }}-D_{\kappa }\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\kappa }^{\vartheta }}+\frac{1}{\tau (\delta ,\varrho )}{D}_{\delta \varrho }^2\frac{\partial {\mathsf{\Theta }}_{1\pi }}{\partial v_{\delta \varrho }^{\vartheta }}=0,\vartheta =\overline{1,k},\pi =\overline{1,m}}$$

$$F_{\pi }^{\iota }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta ,\varrho }\left(t\right),v\left(t\right),t\right)=0,\iota =1,2,\cdots ,r\le n,\pi =1,2,\cdots ,m.$$

Remark 2.

(i)

Next, we formulate the most general Lagrange 1-form that can be used in our context

$${\displaystyle {\mathsf{\Theta }}_{2\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),\omega \left(t\right),t\right)={\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)}$$

$${\displaystyle +{\omega }_{\iota \pi }^{\lambda }\left(t\right)F_{\lambda }^{\iota }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right).}$$

(ii)

The closeness conditions ${\displaystyle D_{\theta }{\mathsf{\Theta }}_{\pi }=D_{\pi }{\mathsf{\Theta }}_{\theta }}$ associated with the Lagrange 1-form for ${\displaystyle {\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)d{t}^{\pi }}$ represent some PDE constraints for our control problem. Moreover, the optimal control problem for ${\displaystyle K\left(b(·),v(·)\right)}$, conditioned by ${\displaystyle D_{\theta }{\mathsf{\Theta }}_{\pi }=D_{\pi }{\mathsf{\Theta }}_{\theta }}$, can be investigated by using

$${\displaystyle {\mathsf{\Theta }}_{3\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),\omega \left(t\right),t\right)={\mathsf{\Theta }}_{\pi }\left(b\left(t\right),b_{\kappa }\left(t\right),b_{\delta \varrho }\left(t\right),v\left(t\right),t\right)}$$

$${\displaystyle +{\omega }_{\pi }^{\theta \lambda }\left(t\right)\left(D_{\theta }{\mathsf{\Theta }}_{\lambda }-D_{\lambda }{\mathsf{\Theta }}_{\theta }\right).}$$

Illustrative example. Find the minimum of the next cost functional

$${\displaystyle K\left(b(·),v(·)\right)={\int }_{C_{0,1}}\left(b^2\left(t\right)+v^2\left(t\right)\right)d{t}^1+\left(b^2\left(t\right)+v^2\left(t\right)\right)d{t}^2}$$

subject to ${\displaystyle b_{t^1}\left(t\right)+b_{t^2}\left(t\right)=0}$ and the boundary conditions ${\displaystyle b(0,0)=0,b(1,1)=0}$, where ${\displaystyle C_{0,1}}$ is a curve of ${\displaystyle C^1}$-class, that is contained in ${[0,1]}^2$, and links ${\displaystyle (0,0)}$ and $(1,1)$.

Solution. By considering that the functional $K\left(b(·),v(·)\right)$ is independent of the path, we obtain

$$b\left(\frac{\partial b}{\partial t^2}-\frac{\partial b}{\partial t^1}\right)=v\left(\frac{\partial v}{\partial t^1}-\frac{\partial v}{\partial t^2}\right).$$

In addition, for the corresponding Lagrange 1-form (see Remark 2), we find

$${\displaystyle {\mathsf{\Theta }}_{11}=b^2\left(t\right)+v^2\left(t\right)+{\omega }_1\left(t\right)\left(b_{t^1}\left(t\right)+b_{t^2}\left(t\right)\right),}$$

$${\mathsf{\Theta }}_{12}=b^2\left(t\right)+v^2\left(t\right)++{\omega }_2\left(t\right)\left(b_{t^1}\left(t\right)+b_{t^2}\left(t\right)\right)$$

and the extreme points are provided by the following PDEs of Euler–Lagrange type

$${\displaystyle 2s-\frac{\partial {\omega }_1}{\partial t^1}-\frac{\partial {\omega }_1}{\partial t^2}=0,2s-\frac{\partial {\omega }_2}{\partial t^1}-\frac{\partial {\omega }_2}{\partial t^2}=0,}$$

$${\displaystyle 2u=0,}$$

$${\displaystyle b_{t^1}\left(t\right)+b_{t^2}\left(t\right)=0.}$$

By direct computation, it follows that ${\displaystyle (b^{*},v^{*})=(0,0)}$ is the optimal point of the considered control problem with PDEs as constraints, and verifies ${\displaystyle \frac{\partial \varphi }{\partial t^1}+\frac{\partial \varphi }{\partial t^2}=0}$, with ${\displaystyle \varphi :={\omega }_1-{\omega }_2}$.

3. Conclusions

We have introduced and analyzed a new type of control problems with PDEs of second-order as constraints. More precisely, we have stated the optimality conditions which are necessary (not sufficient, as well) for a point to be an extreme of the studied control problem with PDEs of second-order as constraints. Since the functionals of curvilinear integral type are useful to compute the mechanical work performed by a variable force, the importance of the considered problem is provided by its applications in mechanics and physics, where second-order PDEs are involved.

As for further development of this paper, we suggest the study of some sufficient conditions of optimality and, also, the associated duality results.