Necessary and sufficient conditions for the second order discrete and differential inclusions with viable constraints (original) (raw)
Related papers
The optimality principle for second-order discrete and discrete-approximate inclusions
2021
This paper deals with the necessary and sufficient conditions of optimality for the Mayer problem of second-order discrete and discrete-approximate inclusions. The main problem is to establish the approximation of second-order viability problems for differential inclusions with endpoint constraints. Thus, as a supplementary problem, we study the discrete approximation problem and give the optimality conditions incorporating the Euler-Lagrange inclusions and distinctive transversality conditions. Locally adjoint mappings (LAM) and equivalence theorems are the fundamental principles of achieving these optimal conditions, one of the most characteristic properties of such approaches with second-order differential inclusions that are specific to the existence of LAMs equivalence relations. Also, a discrete linear model and an example of second-order discrete inclusions in which a set-valued mapping is described by a nonlinear inequality show the applications of these results.
Approximation and optimization of higher order discrete and differential inclusions
Nonlinear Differential Equations and Applications NoDEA, 2014
This paper is mainly concerned with the necessary and sufficient conditions of optimality for Cauchy problem of higher order discrete and differential inclusions. Applying optimality conditions of problems with geometric constraints, for arbitrary higher order (say s-order) discrete inclusions optimality conditions are formulated. Also some special transversality conditions, which are peculiar to problems including third order derivatives are formulated. Formulation of sufficient conditions both for convex and non-convex discrete and differential inclusions are based on the apparatus of locally adjoint mappings. Furthermore, an application of these results is demonstrated by solving the problems with third order linear discrete and differential inclusions.
TURKISH JOURNAL OF MATHEMATICS, 2021
This paper derives the optimality conditions for a Mayer problem with discrete and differential inclusions with viable constraints. Applying necessary and sufficient conditions of problems with geometric constraints, we prove optimality conditions for second order discrete inclusions. Using locally adjoint mapping, we derive Euler-Lagrange form conditions and transversality conditions for the optimality of the discrete approximation problem. Passing to the limit, we establish sufficient conditions to the optimal problem with viable constraints. Conditions ensuring the existence of solutions to the viability problems for differential inclusions of second order have been studied in recent years. However, optimization problems of second-order differential inclusions with viable constraints considered in this paper have not been examined yet. The results presented here are motivated by practices for optimization of various fields as the mass movement model well known in traffic balance and operations research.
ESAIM: COCV, 2020
The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PF C); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PF D). Are proved necessary and sufficient conditions incorporating the Euler-Lagrange inclusion, the Hamil-tonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PF C) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions. Mathematics Subject Classification. 49k20, 49k24, 49J52, 49M25, 90C31.
Optimal control of higher order differential inclusions with functional constraints
ESAIM: Control, Optimisation and Calculus of Variations
The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler-Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and c...
Applicable Analysis, 2020
The paper is devoted to the duality of the Mayer problem for κ-th order viable differential inclusions with endpoint constraints, where κ is an arbitrary natural number. Thus, this paper for constructing the dual problems to viable differential inclusions of any order with endpoint constraints can make a great contribution to the modern development of optimal control theory. For this, using locally conjugate mappings in the form of Euler-Lagrange type inclusions and transversality conditions, sufficient optimality conditions are obtained. It is noteworthy that the Euler-Lagrange type inclusions for both primary and dual problems are 'duality relations'. To demonstrate this approach, some semilinear problems and polyhedral optimization with fourth order differential inclusions are considered. These problems show that sufficient conditions and dual problems can be easily established for problems of a reasonable order. ARTICLE HISTORY
Optimality conditions for higher order polyhedral discrete and differential inclusions
Filomat
The problems considered in this paper are described in polyhedral multi-valued mappings for higher order(s-th) discrete (PDSIs) and differential inclusions (PDFIs). The present paper focuses on the necessary and sufficient conditions of optimality for optimization of these problems. By converting the PDSIs problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type PDSIs and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the PDSIs. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem PDSIs, we reduce this problem to the form of a problem with higher order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher order PDFIs...
Applicable Analysis, 2020
To cite this article: Elimhan N. Mahmudov (2020): Second-order viability problems for differential inclusions with endpoint constraint and duality, Applicable Analysis, ABSTRACT The paper deals with the optimal control of second-order viability problems for differential inclusions with endpoint constraint and duality. Based on the concept of infimal convolution and new approach to convex duality functions, we construct dual problems for discrete and differential inclusions and prove the duality results. It seems that the Euler-Lagrange type inclusions are 'duality relations' for both primary and dual problems. Finally, some special cases show the applicability of the general approach; dual-ity in the control problem with second-order polyhedral DFIs and endpoint constraints defined by a polyhedral cone is considered. ARTICLE HISTORY
THIRD INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2019), 2019
The present paper deals with the free time optimization problem described by third order differential inclusions (P C) with endpoint constraints. In order to construct the optimality conditions for our problem (P C), we obtain the optimality conditions for the discrete-approximation problem associated with the auxiliary differential problem (P A) given by third order convex differential inclusions. Formulation of optimality conditions for problem (P A) plays a substantial role in incorporating the Euler-Lagrange and Hamiltonian type inclusions and moreover by using distinctive t 1-attainability conditions on the initial sets, the sufficient conditions for our main problem (P C) are established.