Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions

1 Introduction

Many extremal problems, including classical optimal control problems, differential games, theory of economic dynamics, and macroeconomic problems, are commonly expressed using multi-valued mappings or variational inclusions, and they constitute an essential component of the mathematical theory of advanced optimization [1–12]. Several papers [13–23] and monographs [4,24,25] have formulated optimal conditions, which are both necessary or sufficient for various optimal control problems involving discrete and continuous processes. The study by Cernea [26] examined a boundary-value problem involving a third-order nonconvex differential inclusion (DFI). It presents several existence conditions achieved through the utilization of the set-valued contraction principle. Andres [27] examined the existence of solutions for a nontraditional target problem involving set-valued flows produced by a vector field represented as a Marchaud map. The proof relies on the generalized Lefschetz trace formula that could be simplified to calculate the sum of local indices under some assumptions on a constraint. In addition to these works, duality, which is a robust and widely utilized technique in convex optimization, plays a crucial role in demonstrating the relationships between the primal problem and its dual counterpart for a variety of reasons. The convexity properties of the primal problem are particularly crucial to this context [5,21,28–30] and related references. In the study by Rockafellar [29], an insightful framework is presented for analyzing optimality conditions for mathematical programming problems through dual programming. The fundamental importance of notions such as the Lagrange function, saddle point, and saddle value is emphasized, with generic cases derived from optimal control, approximation theory, nonlinear optimization, stochastic optimization, and variational calculus. Mordukhovich et al. [5] presented a geometric methodology for variational analysis using Fenchel conjugates as a whole, for convex objects examined within locally convex topological spaces and Banach spaces.

Several scholars have taken into consideration some significant qualitative problems with DFIs such as the existence of optimality [31–39]. In the study by Abbas and Benchohra [39], Ulam’s type stability is studied for the Darboux problem of partial fractional DFIs when the right-hand side of inclusion is nonconvex-valued. Their outcomes depend on the utilization of the Covitz-Nadler fixed point theorem and a fractional variant of Gronwall’s inequality. Furthermore, substantial qualitative research is currently going forward on the subject of higher-order differential inclusions (HODIs) [40–44]. In the study by Bartuzel and Fryszkowski [43], a suggested version of the Filippov lemma is proposed for third-order DFIs that are associated with constraints on boundary conditions. The existence of solutions of kth-order DFIs is studied under various boundary conditions in the study by Frigon and Kaczynski [44].

Boundary-value problems have significant implications in fields ranging from physics and engineering to economics and biology, making them a vital area of research in applied mathematics. Azzam et al. [45] examined the issue of three boundary conditions for second-order DIs. In the work by Benchohra et al. [46], existence problems for DFIs with boundary conditions are studied. The study by Mahmudov [11] is about to formulating optimality conditions for Darboux-type DFI with constraints on the boundary by utilizing adjoint partial DFIs. Matzakos and Papageorgiou [47] investigated a periodic evolution equation governed by a time-dependent subdifferential and featuring a multivalued forcing term. By employing a fixed-point theorem for pseudo-acyclic multifunctions, the existence of periodic trajectories is established. This methodology entails an examination of the solution set’s structure for the Cauchy problem addressed in the work.

In addition to the Lagrangian and Fenchel dualities, the notion of infimal convolution of convex functions has been effectively utilized in some papers [5,21,29]. Mahmudov and Mardanov [21] explored the optimality conditions for the k th-order DFIs and their corresponding duality approach. As far as we know, there have been such works [5,29] dedicated to the dual problems of the first-order DFIs. Some duality results by employing the dual operations of addition and infimal convolution of convex functions are obtained. It is worth noting that the challenges encountered in this approach stem from constructing duality framework for both discrete and discrete approximate problems.

In the current work, we discuss one of the most challenging and captivating optimization problems with higher-order DFIs and special boundary-value constraints. Indeed, the challenge in higher-order DFIs lies more in formulating the optimality conditions, which are, in general, Euler-Lagrange-type higher-order adjoint inclusions and transversality conditions. We use the so-called Mahmudov’s adjoint inclusions to obtain sufficient conditions for the k th-order DFIs instead of Euler-Langrange inclusions, which are used to obtain sufficient conditions for the first-order DFIs. It is evident that when k = 1 , Mahmudov’s and Euler-Lagrange adjoint inclusions coincide. Among the published works, no research has been exclusively dedicated to sufficient conditions for the k th-order DFIs. Therefore, we aim to address this issue, which constitutes a novelty of our specific boundary-value problem with higher-order DFIs. The presented problem is convex; however, without loss of generality, we can extend our results to the non-convex case. Moreover, this article is devoted to the duality of k th-order ordinary DFIs with special BVPs. Then, we applied our duality results to linear and polyhedral optimal control problems. The results for constructed dual problems are also a novelty to optimization literature. The structure of the work is outlined in the following order.

In Section 2, we summarize the basic definitions, concepts, and fundamental outcomes presented in Mahmudov’s book [4]. In particular, the Hamiltonian function and the sets of argmaximum of a multi-valued mapping together with LAM are introduced. Our main problem (PHC) formulated by higher-order DFIs and special boundary-value constraints is defined. Moreover, discrete-approximate problem related to our problem (PHC) is expressed.

In Section 3, we examine the optimization problem (PHC) for an arbitrary kth-order DFI with special boundary-value constraints. The sufficient conditions for BVPs with kth-order DFIs are obtained using convex and nonsmooth analysis in the form of general Mahmudov’s inclusion and with its corresponding transversality conditions. The higher-order adjoint inclusions for a closed multi-valued mapping F could be reformulated suitably with regard to the Hamiltonian function. The relation of LAMs with the subdifferential of Hamiltonian function and arg-maximum sets is essential in this simplification. Note that the main difficulty in obtaining sufficient conditions for optimality in our optimization problem is closely associated with the generalization of the Lagrange functional in cost function and DFI and the existence of special boundary constraints. We have not conducted an analysis of the discrete-approximate problem involving higher-order difference operators in this study.

In Section 4, we apply the results from Section 3 to variational problem and find out the Euler-Poisson equation, which is well known in the calculus of variations. It should be noted that the Euler-Poisson equation is obtained using the adjoint Mahmudov inclusion. Then, the higher-order “linear” optimal control problem is considered as a second application because of its applicability to control problems in engineering and science. Finally, the higher-order Euler-Lagrange-type adjoint differential equation, Weierstrass-Pontryagin maximum principle, and transversality conditions are proved for the arbitrary-order semi-linear problem (PHL).

In Section 5, we construct the dual problem (PHC*) and give an example from linear optimal control theory. The dual problem is generated without computing discrete and discrete-approximation problems. It is proved that there is no gap between the optimal values of the primal convex (PHC) and dual concave (PHC*) problems. Then, the dual problem (PHL*) by utilizing our dual theorem is formulated for the primal problem (PHL).

In Section 6, sufficient conditions and duality theorem are calculated for third-order polyhedral DFIs with BVP.

2 Preliminaries and problem statements

We summarize the essential ideas, definitions, and notions from Mahmudov’s monograph [4]. Let us assume that R n is an n -dimensional Euclidean space, ( x , v ) is a pair of elements x , v ∈ R n , and ⟨ x , v ⟩ is an inner product of x and v . Suppose F : R k n ⇉ R n is a multi-valued (set-valued) mapping from R k n to P ( R n ) . Then, F is said to be convex if the graph of F , which is gph F = { ( x , v 1 , … , v k − 1 , v k ) : v k ∈ F ( x , v 1 , … , v k − 1 ) } is a convex subset in R ( k + 1 ) n . When graph of a multi-valued function F is a closed set in R ( k + 1 ) n , the mapping F can be said to be closed. When F ( x , v 1 , … , v k − 1 ) is a convex set for each ( x , v 1 , … , v k − 1 ) ∈ dom F = { ( x , v 1 , … , v k − 1 ) : F ( x , v 1 , … , v k − 1 ) ≠ ∅ } , F can be said to be convex-valued. The Hamiltonian function and argmaximum set for a multi-valued mapping F are described by the following relations:

respectively. When F is convex multi-valued mapping, we place H F ( x , v 1 , … , v k − 1 , v k * ) = − ∞ if F ( x , v 1 , … , v k − 1 ) = ∅ . Assume that int A is the interior of the set A ⊂ R ( k + 1 ) n and ri A denotes the relative interior of the set A , which is defined as the set of interior points of A in terms of its affine hull Aff A .

A convex cone K A ( x , v 1 , … , v k − 1 ) is a cone of tangent directions at a point ( x , v 1 , … , v k − 1 ) ∈ A ⊂ R ( k + 1 ) n if from ( x ¯ , v ¯ 1 , … , v ¯ k − 1 ) ∈ K A ( x , v 1 , … , v k − 1 ) , it follows that ( x ¯ , v ¯ 1 , … , v ¯ k − 1 ) is a tangent vector at point ( x , v 1 , … , v k − 1 ) ∈ A . Moreover, let us denote K A * ( ⋅ ) as the dual cone to cone K A ( ⋅ ) .

A function g = g ( x , v 1 , … , v k − 1 ) is said to be a proper function when it does not take the value of − ∞ and is not identical to + ∞ . That g is proper function is necessary and sufficient for that dom g ≠ ∅ and g ( x , v 1 , … , v k − 1 ) is finite for ( x , v 1 , … , v k − 1 ) ∈ dom g = { ( x , v 1 , … , v k − 1 ) : g ( x , v 1 , … , v k − 1 ) < + ∞ } .

A function g = g ( x , v 1 , … , v k − 1 ) is referred to as a closed function when the epigraph of g ( ⋅ ) , which is epi g = { ( η , x , v 1 , … , v k − 1 ) : η ≥ g ( x , v 1 , … , v k − 1 ) } , is a closed set.

The function g * ( x * , v 1 * , … , v k − 1 * ) described as follows is referred to the conjugate of g ( ⋅ ) :

Support function for the set A is denoted as W A ( x * ) and defined as follows

It is well known that conjugate functions are always convex and closed regardless of the original function. Let us indicate

We can conclude easily the relation between M F and H F as follows:

It is straightforward to observe that the function has the following property:

represents a support function with a negative sign. Moreover, this implies for a fixed v * :

In other words, M F is the conjugate function of H F ( ⋅ , v * ) with a negative sign.

The multi-valued mapping F * ( ⋅ , x , v 1 , … , v k − 1 , v k ) : R n ⇉ R k n can be characterized as follows when F is a convex multi-valued mapping:

is referred to locally adjoint mapping (LAM) of the multi-valued F at a point ( x 0 , v 1 0 , … , v k − 1 0 , v k 0 ) ∈ gph F . By Lemma 2.6 [4, p. 64], it is noteworthy to see that ( x * , v 1 * , … , v k − 1 * ) is an element of the LAM F * in the case that

A multi-valued mapping for “nonconvex” multi-valued mapping F at a point ( x 0 , v 1 0 , … , v k − 1 0 , v k 0 ) ∈ gph F defined by

is labeled as the LAM. If F is convex mapping, the Hamiltonian H F ( ⋅ , v k * ) is clearly concave; therefore, this description of LAM coincides with the earlier description of LAM in that case. Indeed, the closely related concept of coderivative (Frechet) generated by normal cones of multi-valued mappings with their graphs has been introduced by Mordukhovich [24]. This notion of derivative is essentially different from LAM in the case the multi-valued mapping F is not convex. Nevertheless, if the multi-valued mapping F is smooth and convex, then the coderivative notion and LAM are equivalent to each other.

In Section 3, we address the Lagrange problem of HODIs with special boundary conditions:

where F ( ⋅ , t ) : R n k ⇉ R n is a convex multi-valued mapping, g ( ⋅ , t ) : R n k → R 1 is a proper convex function, and μ s ∈ R n ( s = 0 , … , k − 1 ) are constant vectors, k is a random natural number, and T is a random positive real number. It can be seen easily that the problem (PHC) is convex since F ( ⋅ , t ) is a convex multi-valued mapping and g ( ⋅ , t ) is a convex proper function. In this study, we have to find an admissible trajectory arc x ˜ ( t ) of the problem (PHC) that satisfies condition (2) almost everywhere (a.e.) on a time interval [ 0 , T ] , and the boundary conditions (3) minimizing the so-called Lagrange cost functional J [ x ( ⋅ ) ] . It should be mentioned when μ s = 0 , s = 0 , 1 , … , k − 1 , conditions (3) are periodic boundary conditions. The notion of a solution of the problem for k th-order DFIs (2) should be clarified. Assume that A C j ( [ 0 , T ] , R n ) is the space of j times differentiable functions x ( ⋅ ) : [ 0 , T ] → R n , where j th-order derivative x ( j ) ( ⋅ ) ≡ d ( j ) x ( ⋅ ) ∕ d t j ( j = 1 , … , k ) is absolutely continuous. Moreover, we can suppose that L 1 ( [ 0 , T ] , R n ) is the Banach space of integrable functions u ( ⋅ ) : [ 0 , T ] → R n endowed with the norm ‖ u ( ⋅ ) ‖ 1 = ∫ 0 T ∣ u ( t ) ∣ d t in the Lebesgue sense. A function x ( ⋅ ) ∈ A C k − 1 ( [ 0 , T ] , R n ) is a feasible solution of a problems (2) and (3) if there exists an integrable function v ( t ) ∈ F ( x ( t ) , x ′ ( t ) , … , x ( k − 1 ) ( t ) , t ) , a.e. t ∈ [ 0 , T ] such that x ( k ) ( t ) = v ( t ) a.e. t ∈ [ 0 , T ] and x ( ⋅ ) satisfies the boundary conditions (3). It should be noted that having the boundary conditions as a constraint x ( s ) ( 0 ) − x ( s ) ( T ) = μ s , s = 0 , 1 , … , k − 1 are especially substantial for applied scientists. For example, in the book of Aubin and Cellina [1], first order DFIs and their various applications are introduced in detail. These applications frequently emerge in the field of optimal control, mathematical finance, classical mechanics, designing optimal or stabilizing feedback, as well as aerospace engineering, anti-vibration, burning in rocket motors, and biophysics.

Note that to obtain sufficient conditions for optimality of the higher-order Langrange problem with special boundary conditions, we can use the discrete-approximate method. Hence, our problem (PHC) is replaced by the subsequent k th-order discrete approximate problem (PHDA):

Here, the k th-order finite difference operator can be described in this fashion:

The main idea for this method is to transform the continuous problem (PHC) to discrete problem (PHDA) by the finite difference approximation of differential operators. Then, optimality conditions for discrete inclusion need to be derived and applied to discrete-approximation problem. Subsequently, taking limit as h → 0 , we can construct the optimality conditions for continuous problem (PHC). It is obvious that these procedures are rather complicated due to high-order derivatives. Although the method of discrete-approximation is a brilliant technique for the investigation of optimality conditions of our problem, we omit it to avoid long and tedious calculations. Therefore, we will approach to the problem (PHC) in a different way so as to prove theorems in this article.

3 Sufficient conditions for problem (PHC)

We initiate our consideration with sufficient conditions for the addressed problem (PHC). To begin, we relate the problem (PHC) to the succeeding kth-order Euler-Lagrange-type DFI or the so-called Mahmudov’s inclusion:

1. ( − 1 ) k d k x * ( t ) d t k + d φ k − 1 * ( t ) d t , φ k − 1 * ( t ) + d φ k − 2 * ( t ) d t , … , φ 2 * ( t ) + d φ 1 * ( t ) d t , φ 1 * ( t ) ∈ F * ( x * ( t ) ; ( x ˜ ( t ) , x ˜ ′ ( t ) , … , x ˜ ( k ) ( t ) ) , t ) − ∂ z g ( x ˜ ( t ) , x ˜ ′ ( t ) , … , x ˜ ( k − 1 ) ( t ) , t ) , a.e. t ∈ [ 0 , T ] , where x ˜ ( t ) is a feasible solution of the problem ( PHC ) , z = ( x , v 1 , … , v k − 1 ) , x * ( t ) and φ * ( t ) are the so-called adjoint variables and the transversality conditions at the points t = 0 and t = T :

2. x * ( s ) ( 0 ) = x * ( s ) ( T ) ; s = 0 , 1 , … , k − 1 .

3. φ j * ( 0 ) = φ j * ( T ) ; j = 1 , 2 , … , k − 1 .

Additionally, concerning the argmaximum set, we will need an additional condition stipulating LAM F * ( ⋅ , t ) ≠ ∅ at a specified point:

1. d k x ˜ ( t ) d t k ∈ F A ( x ˜ ( t ) , x ˜ ′ ( t ) , … , x ˜ ( k − 1 ) ( t ) ; x * ( t ) , t ) , a.e. t ∈ [ 0 , T ] .