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1

Feedback design of differential equations of reconstruction for second-order distributed parameter systems

Maksimov V. I., Mordukhovich B. S.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 3

467--475

EN

The paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system.

2

Optimal control of delay-differential inclusions with multivalued initial conditions in infinite dimensions

Mordukhovich B. S., Wang D., Wang L.

Control and Cybernetics

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2008

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Vol. 37, no 2

393-428

EN

This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with infinite-dimensional state spaces, endpoints constraints, and multivalued initial conditions. To the best of our knowledge, problems of this type have not been considered in the literature, except for some particular cases when either the state space is finite-dimensional or there is no delay in the dynamics. We develop the method of discrete approximations to derive necessary optimality conditions in the extended Euler-Lagrange form by using advanced tools of variational analysis and generalized differentiation in infinite dimensions. This method consists of the three major parts: (a) constructing a well-posed sequence of discrete-time problems that approximate in an appropriate sense the original continuous-time problem of dynamic optimization; (b) deriving necessary optimality conditions for the approximating discrete-time problems by reducing them to infinite-dimensional problems of mathematical programming and employing then generalized differential calculus; (c) passing finally to the limit in the obtained results for discrete approximations to establish necessary conditions for the given optimal solutions to the original problem. This method is fully realized in the delay-differential systems under consideration.

3

Variational principles for set-valued mappings with applications to multiobjective optimization

Bao T. Q., Mordukhovich B. S.

Control and Cybernetics

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2007

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Vol. 36, no 3

531-562

EN

This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which-being certainly of independent interest-are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang, formulated via an appropriate subdifferential notion for set-valued mappings with values in partially ordered spaces. Based on these variational principles and corresponding tools of generalized differentiation, we derive new conditions of the coercivity and Palais-Smale types ensuring the existence of optimal solutions to set-valued optimization problems with noncompact feasible sets in infinite dimensions and then obtain necessary optimality and suboptimality conditions for nonsmooth multiobjective optimization problems with general constraints, which are new in both finite-dimensional and infinite-dimensional settings.

4

Optimal control of semilinear evolution inclusions via discrete approximations

Mordukhovich B. S., Wang D.

Control and Cybernetics

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2005

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Vol. 34, no 3

849-870

EN

This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for constrained continuous-time evolution inclusions by passing to the limit from discrete approximations.

5

Optimal control of constrained delay-differential inclusions with multivalued initial conditions

Mordukhovich B. S., Wang L.

Control and Cybernetics

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2003

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Vol. 32, no 3

585-609

EN

This paper studies a general optimal control problem for nonconvex delay-differential inclusions with endpoint constraints. In contrast to previous publications on this topic, we incorporate time-dependent set constraints on the initial interval, which are specific for systems with delays and provide an additional source for optimization. Our variational analysis is based on well-posed discrete approximations of constrained delay-differential inclusions by a family of time-delayed systems with discrete dynamics and perturbed constraints. Using convergence results for discrete approximations and advanced tools of nonsmooth variational analysis, we derive necessary optimality conditions for constrained delay-differential inclusions in both Euler-Lagrange and Hamiltonian forms involving nonconvex generalized differential constructions for nonsmooth functions, sets, and set-valued mappings.

6

Mixed coderivatives of set-valued mappings in variational analysis

Mordukhovich B. S., Shao Y.

Journal of Applied Analysis

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1998

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Vol. 4, nr 2

269-294

EN

We consider a refined coderivative construction for nonsmooth and set-valued mappings between Banach spaces. This limiting mixed coderivative is different from "normal" coderiva-tives generated by normal cones/subdifferentials and turns out to be useful for studying some basic propertiers in variational analysis particularly related to Lipschitzian stability. We develop a strong calculus for this coderivative important for various applications.