Convex Sets and Functions

Mahmudov, Elimhan N.

doi:10.1016/b978-0-12-388428-2.00001-1

Cited by 9 publications

(33 citation statements)

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“…is concave and the latter definition of LAM coincide with the previous definition of LAM (Theorem 2.1 [13]). Note that prior to the LAM the notion of coderivative has been introduced for set-valued mappings in terms of the basic normal cone to their graphs by Mordukhovich [24] and for the smooth and convex maps the two notions are equivalent.…”

Section: Elimhan N Mahmudovsupporting

confidence: 55%

“…The present paper is ordered in the following manner. In Section 2 are given the necessary facts and supplementary results from the book of Mahmudov [13]; Hamiltonian function and locally adjoint mapping (LAM) are introduced and the problem with initial point constraints for PLDO's governed by time-dependent set-valued mapping are formulated. In Section 3, we present the main results; on the basis of transversality conditions at the endpoints of the considered time interval are proved the sufficient conditions of optimality for differential inclusions with PLDO's and with initial point constraints.…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

“…Preliminaries and problems statements. In this section we recall the key notions of set-valued mappings from the book [13]; let R n be a n-dimensional Euclidean space, x, v be an inner product of elements x, v ∈ R n , (x, v) be a pair of x, v. Let F : R n ⇒ R n be a set-valued mapping from R n into the set of subsets of R n . Therefore F R 3n ⇒ R n is a convex set-valued mapping, if is a convex set-valued mapping, if its graph gphF = {(x, v) : v ∈ F (x)} is a convex subset of R 2n .…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

“…Thus by using the method of approximation [13], [14], [24] we can establish necessary and sufficient conditions for rather complicated. Then by passing to the limit in necessary and sufficient conditions of this problem as h → 0, we can construct the optimality condition for Mayer problem (P V ) described by higher order differential inclusions with PLDOs and with initial point constraints.…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

“…Proof. It is not hard to see that by using the Theorem 2.1 [13] in term of Hamiltonian function from the condition (i) we derive the important inequality…”

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 Mahmudovsupporting

confidence: 55%

Section: Elimhan N Mahmudovmentioning

confidence: 99%

Section: Elimhan N Mahmudovmentioning

confidence: 99%

Section: Elimhan N Mahmudovmentioning

confidence: 99%

“…Proof. It is not hard to see that by using the Theorem 2.1 [13] in term of Hamiltonian function from the condition (i) we derive the important inequality…”

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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Optimal Preview Control for Discrete-Time Systems in Multirate Output Sampling

Liao

Guo²

2016

Mathematical Problems in Engineering

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This paper studies the disturbance preview optimal control problem for discrete-time systems with multirate output sampling. By constructing the error system and using the discrete lifting technique, we reduce the multirate preview control problem to a single-rate one for a formal augmented system. Then, applying preview control theory, the optimal preview control law of the augmented error system is obtained. Meanwhile, we introduce a discrete integrator to eliminate the static error. Then we study a method to design a controller with preview action for the original system. And the existence conditions of the controller are also discussed in detail. Finally, numerical simulation is included to illustrate the effectiveness of the proposed method.

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

Mahmudov¹

2018

Evolution Equations &Amp; Control Theory

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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.

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Convex Sets and Functions

Cited by 9 publications

References 0 publications

Optimal control of evolution differential inclusions with polynomial linear differential operators

Optimal control of evolution differential inclusions with polynomial linear differential operators

Optimal Preview Control for Discrete-Time Systems in Multirate Output Sampling

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

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