Recent Advances of Constrained Variational Problems Involving Second-Order Partial Derivatives: A Review

: This paper comprehensively reviews the nonlinear dynamics given by some classes of constrained control problems which involve second-order partial derivatives. Speciﬁcally, necessary optimality conditions are formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and the associated variational inequalities are considered in the present review paper.

This review article is structured as follows. Section 2 introduces the second-order PDEconstrained optimal control problem under study (see Theorem 1). This result formulates the necessary conditions of optimality for the considered PDE-constrained optimization problem. Section 3 states the associated necessary optimality conditions for a new class of isoperimetric constrained control problems governed by multiple and curvilinear integrals. In Section 4, by using the pseudomonotonicity, hemicontinuity, and monotonicity of the considered integral functionals, we present the well-posedness of some variational inequality problems determined by partial derivatives of a second-order. Section 5 formulates some very important open problems to be investigated in the near future. Section 6 contains the conclusions of the paper.
as a hyper-parallelepiped determined by the diagonally opposite points t 0 , t 1 ∈ R m + . Moreover, we assume that the previous multi-time controlled Lagrangians of second order determine a closed controlled Lagrange 1-form (see summation over the repeated indices), which provides the following curvilinear integral functional: where Υ t 0 ,t 1 is a smooth curve, included in Λ t 0 ,t 1 , joining t 0 , t 1 ∈ R m + . Second-order PDE-constrained control problem. Find the pair (b * , u * ) that minimizes the aforementioned controlled path-independent curvilinear integral functional Equation (1), among all the pair functions (b, u) satisfying and the partial speed-acceleration constraints: , u(t), t = 0, a = 1, 2, · · · , r ≤ n, ζ = 1, 2, · · · , m.
In order to investigate the above controlled optimization problem in Equation (1), associated with the aforementioned partial speed-acceleration constraints, we introduce the Lagrange multiplier p = (p a (t)) and build new multi-time-controlled second-order Lagrangians (see summation over the repeated indices): , t , ζ = 1, m, which change the initial controlled optimization problem (with second-order PDE constraints) into a partial speed-acceleration, unconstrained, controlled optimization problem: In accordance with Lagrange theory, an extreme point of Equation (1) is found among the extreme points of Equation (2).
The following theorem represents the main result of this section (see Treanţȃ [12]). It establishes the necessary conditions of optimality associated with the considered secondorder PDE-constrained control problem.

Isoperimetric Constrained Controlled Optimization Problem
In this section, we use similar notations as in the previous section. We consider as a hyper-parallelepiped generated by the diagonally opposite points t 0 , t 1 ∈ R m + . Isoperimetric constrained control problem. Find the pair (b * , u * ) that minimizes the following multiple integral functional: and the isoperimetric constraints (that is, constant level sets of some functionals) formulated as follows.
In order to investigate the above controlled optimization problem in Equation (3), associated with the aforementioned isoperimetric constraints, we introduce the curve Υ t 0 ,t ⊂ Υ t 0 ,t 1 and the auxiliary variables: which satisfy y a (t 0 ) = 0, y a (t 1 ) = l a . Consequently, the functions y a fulfill the next first-order PDEs: Considering the Lagrange multiplier p = p ζ a (t) and by denoting y = (y a (t)), we introduce a new multi-time-controlled Lagrangian of second order: ∂y a ∂t ζ (t) that changes the initial control problem into an unconstrained control problem min b(·), u(·), y(·), p(·) Λ t 0 ,t 1 In accordance with Lagrange theory, an extreme point of Equation (3) is found among the extreme points of Equation (4).
The following theorem (see Treanţȃ and Ahmad [13]) establishes the necessary conditions of optimality associated with the considered isoperimetric constrained control problem.

Remark 3 (Treanţȃ and Ahmad [13]). The system of Euler-Lagrange PDEs given in Theorem 2 becomes
In consequence, the Lagrange matrix multiplier p has null total divergence. Moreover, it is well determined only if the optimal solution is not an extreme for at least one of the functionals

Well-Posedness of Some Variational Inequalities Involving Second-Order Partial Derivatives
In the following, in accordance with Treanţȃ [14][15][16], we consider: Λ s 1 ,s 2 as a compact set in R m ; Λ s 1 ,s 2 s = (s ζ ), ζ = 1, m as a multi-variate evolution parameter; Λ s 1 ,s 2 ⊃ Υ as a piecewise differentiable curve that links the points s 1 = (s 1 1 , . . . , s m 1 ), s 2 = (s 1 2 , . . . , s m 2 ) in Λ s 1 ,s 2 ; B as the space of C 4 -class state functions b : Λ s 1 ,s 2 → R n ; and b κ := ∂b ∂s κ , b αβ := ∂ 2 b ∂s α ∂s β denote the partial speed and partial acceleration, respectively. In addition, let U be the space of C 1 -class control functions u : Λ s 1 ,s 2 → R k and assume that B × U is a (nonempty) convex and closed subset of B × U, equipped with Let J 2 (R m , R n ) be the jet bundle of the second order of R m and R n . Assume that the Lagrangians w ζ : which gives the following integral functional: In order to state the problem under study, we introduce the Saunders's multi-index (Saunders [32]). Now, we introduce the variational problem: where D κ := ∂ ∂s κ is the total derivative operator, Let Ω be the feasible solution set of (5): Assumption 1. The next working hypothesis is assumed: as a total exact differential, with G(s 1 ) = G(s 2 ).
According to Equation (6) and considering the notion of monotonicity associated with variational inequalities, we formulate (see Treanţȃ et al. [14]) the monotonicity and pseudomonotonicity for W.
By using Usman and Khan [33], we introduce the following definition.
Lemma 1 (Treanţȃ et al. [14]). Let the functional W be hemicontinuous and pseudomonotone on Furthermore, according to Treanţȃ et al. [14], we present two well-posedness results associated with the considered variational inequality problem involving second-order PDEs. Definition 4. The sequence {(b n , u n )} ∈ B × U is called an approximating sequence of Equation (5) if there exists a sequence of positive real numbers σ n → 0 as n → ∞, such that: Definition 5. The problem Equation (5) is called well-posed if: (i) The problem in Equation (5) has one solution (b 0 , u 0 ); (ii) Each approximating sequence of Equation (5) converges to (b 0 , u 0 ). The approximating solution set of Equation (5) is given as follows:

Remark 4.
We have: Ω = Ω σ , when σ = 0 and Ω ⊆ Ω σ , ∀σ > 0. Furthermore, for a set P, the diameter of P is defined as follows Theorem 3 (Treanţȃ et al. [14]). Let the functional W be hemicontinuous and monotone on B × U. The problem Equation (5) is well-posed if and only if: Theorem 4 (Treanţȃ et al. [14]). Let the functional W be hemicontinuous and monotone on B × U. Then, Equation (5) is well-posed if and only if it has one solution.

Open Problem
As in the previous sections, we start with T as a compact set in R m and T ζ = (ζ β ), β = 1, m, as a multi-variable. Let T ⊃ C : ζ = ζ(ς), ς ∈ [p, q] be a (piecewise) differentiable curve joining the following two fixed points ζ 1 = (ζ 1 1 , . . . , ζ m 1 ), ζ 2 = (ζ 1 2 , . . . , ζ m 2 ) in T . In addition, we consider Λ as the space of (piecewise) smooth state functions σ : T → R n and Ω as the space of control functions η : T → R k , which are considered to be piecewise continuous. Moreover, on the product space Λ × Ω, we consider the scalar product: together with the norm induced by it.
For examples of invex and/or pseudoinvex curvilinear integral functionals, the reader can consult Treanţȃ [17].