Optimal control of second order delay-discrete and delay-differential inclusions with state constraints

Abstract

The present paper studies a new class of problems of optimal control theory with state constraints and second order delay-discrete (DSIs) and delay-differential inclusions (DFIs). The basic approach to solving this problem is based on the discretization method. Thus under the regularity condition the necessary and sufficient conditions of optimality for problems with second order delay-discrete and delay-approximate DSIs are investigated. Then by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions the sufficient conditions of optimality for delay-DFIs, including the peculiar transversality ones, are proved. Here our main idea is the use of equivalence relations for subdifferentials of Hamiltonian functions and locally adjoint mappings (LAMs), which allow us to make a bridge between the basic optimality conditions of second order delay-DSIs and delaydiscrete-approximate problems. In particular, applications of these results to the second order semilinear optimal control problem are illustrated as well as the optimality conditions for non-delayed problems are derived.

Optimal Control of Evolution Differential Inclusions with Polynomial Linear Differential Operators

“…Therefore, sinceũt ðÞis a piecewise constant function, having not more than two intervals of constancy, we have either the cases Eqs. (21) and (22) or the case Eq. (23).…”

“…The optimization of higher order differential inclusions was first developed by Mahmudov in [14][15][16][17][18][19][20][21]. Since then this problem has attracted many author's attentions (see [22] and their references).…”

“…Thus with the use of Euler-Lagrange and Hamiltonian type of inclusions and transversal conditions on the "initial" sets, the sufficient conditions are formulated. The paper [21] studies a new class of problems of optimal control theory with state constraints and second-order delay discrete and delay differential inclusions. Under the "regularity" condition by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions, the sufficient conditions of optimality for delay DFIs, including the peculiar transversality ones, are proved.…”

“…Here, in the case Eq. (21),ũt ðÞ¼1, t ∈ 0, 1 ½ , and so from Eq. (26), we havex 1 1 ðÞ¼ 0:5 þ 1 ¼ 1:5;x 0 1 1 ðÞ¼2.…”

Optimal control of evolution differential inclusions with polynomial linear differential operators

“…The optimization of higher order differential inclusions was first developed by Mahmudov in [16]- [23]. Since then this problem has attracted many author's attentions (see [5] and their references).…”