Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

Mahmudov, Elimhan N.

doi:10.1007/s10957-018-1260-2

Cited by 35 publications

(26 citation statements)

References 21 publications

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“…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).…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

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Optimal control of evolution differential inclusions with polynomial linear differential operators

Mahmudov¹

2019

Evolution Equations &Amp; Control Theory

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In this paper we have introduced a new class of problems of optimal control theory with differential inclusions described by polynomial linear differential operators. Consequently, there arises a rather complicated problem with simultaneous determination of the polynomial linear differential operators with variable coefficients and a Mayer functional depending on high order derivatives. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and transversality conditions are derived. Formulation of the transversality conditions at the endpoints of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with polynomial linear differential operators is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem and the Weierstrass-Pontryagin maximum principle is obtained. Then the optimality conditions are derived for second order convex differential inclusions with convex endpoint constraints.

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Section: Elimhan N Mahmudovmentioning

confidence: 99%

“…Since then this problem has attracted many author's attentions (see [5] and their references). In the papers [16], [20] are studied a new class of problems of optimal control theory with Sturm-Liouville type differential inclusions involving second and fourth orders linear self-adjoint differential operators.…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

Optimal control of evolution differential inclusions with polynomial linear differential operators

Mahmudov¹

2019

Evolution Equations &Amp; Control Theory

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“…By using the equivalence theorem concerning the subdifferential calculus and approximating method, a necessary and sufficient condition for the discrete-approximation problem with inequality constraint was established. In [15], the author studied a new class of problems of optimal control theory with Sturm-Liouville type differential inclusions involving second order linear self-adjoint differential operators. By using the discretization method guaranteeing transition to the continuous problem, the discrete and discrete-approximate inclusions were investigated.…”

Section: Introduction-preliminariesmentioning

confidence: 99%

Optimization of the Bolza problem with second order differential inequalities,

2018

J. Nonlinear Funct. Anal.

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This paper concerns optimization of the Bolza problem with convex and nonconvex second order discrete and differential state variable inequality constraints. Necessary and sufficient conditions of optimality for second order discrete and differential inequalities are derived. According to proposed discretization method, the problem with discrete-approximation inequalities is investigated. Equivalence theorems for subdifferential inclusions are basic tools in the study of optimality conditions for continuous problems. This approach plays a much more important role in the derivation of second order adjoint discrete and differential inequality constraints generated by given inequality constraints. A numerical example is presented to illustrate the theoretical result.

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“…In spite of the presence of the above mentioned qualitative studies, optimal control of problems with the higher order differential inclusions has not been studied in the literature; up to our best knowledge, there a few papers of Mahmudov [18][19][20][21][22][23][24] devoted to such studies. The paper [19] concerns itself with the second order polyhedral optimization described by ordinary discrete and differential inclusions.…”

mentioning

confidence: 99%

“…Necessary and sufficient conditions of optimality including distinctive transversality condition are proved in the form of Euler-Lagrange inclusions. The paper [23] studies a new class of problems of optimal control theory with Sturm-Liouville type differential inclusions involving second order linear self-adjoint differential operators. Necessary and sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and "transversality" conditions, are derived.…”

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confidence: 99%

Optimal control of higher order differential inclusions with functional constraints

Mahmudov

2020

ESAIM: COCV

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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 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 (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.

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Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

Cited by 35 publications

References 21 publications

Optimal control of evolution differential inclusions with polynomial linear differential operators

Optimal control of evolution differential inclusions with polynomial linear differential operators

Optimization of the Bolza problem with second order differential inequalities,

Optimal control of higher order differential inclusions with functional constraints

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