# On the Duality Theory for Problems with Higher Order Differential Inclusions

## TL;DR

This paper develops a duality theory for higher-order differential inclusions in optimal control, deriving Euler-Lagrange type conditions and demonstrating duality through specific examples, advancing the mathematical foundation of control problems.

## Contribution

It introduces a general duality framework for k-th order differential inclusions, including optimality conditions and duality theorems, for any order in control theory.

## Key findings

- Derived Euler-Lagrange type optimality conditions for higher-order inclusions
- Established duality relations and theorems for second order polyhedral inclusions
- Demonstrated the approach with semilinear problems of arbitrary order

## Abstract

This paper on the whole concerns with the duality of Mayer problem for k-th order differential inclusions, where k is an arbitrary natural number. Thus, this work for constructing the dual problems to differential inclusions of any order can make a great contribution to the modern development of optimal control theory. To this end in the form of Euler-Lagrange type inclusions and transversality conditions the sufficient optimality conditions are derived. The principal idea of obtaining optimal conditions is locally adjoint mappings. It appears that the Euler-Lagrange type inclusions for both primary and dual problems are "duality relations". To demonstrate this approach, some semilinear problems with k-th order differential inclusions are considered. Also, the optimality conditions and the duality theorem in problems with second order polyhedral differential inclusions are proved. These problems show that maximization in the dual problems are realized over the set of solutions of the Euler-Lagrange type differential inclusions/equations.

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Source: https://tomesphere.com/paper/1906.07721