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1
Content available Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality
100%
Ahmed N.
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tom 35
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nr 2
165-195
EN
In this paper we consider controlled McKean-Vlasov stochastic evolution equations on Hilbert spaces. We prove existence and uniqueness of solutions and regularity properties thereof. We use relaxed controls, adapted to a current of sub-sigma algebras generated by observable processes, and taking values from a Polish space. We introduce an appropriate topology based on weak star convergence. We prove continuous dependence of solutions on controls with respect to appropriate topologies. Theses results are then used to prove existence of optimal controls for Bolza problems. Then we develop the necessary conditions of optimality based on semi-martingale representation theory on Hilbert spaces. Next we show that the adjoint processes arising from the necessary conditions optimality can be constructed from the solution of certain BSDE.
2
Content available A general class of McKean-Vlasov stochastic evolution equations driven by Brownian motion and Lèvy process and controlled by Lèvy measure
100%
Ahmed N.
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tom 36
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nr 2
181-206
EN
In this paper we consider McKean-Vlasov stochastic evolution equations on Hilbert spaces driven by Brownian motion and L`evy process and controlled by L`evy measures. We prove existence and uniqueness of solutions and regularity properties thereof. We consider weak topology on the space of bounded Le´vy measures on infinite dimensional Hilbert space and prove continuous dependence of solutions with respect to the Le´vy measure. Then considering a certain class of Le´vy measures on infinite as well as finite dimensional Hilbert spaces, as relaxed controls, we prove existence of optimal controls for Bolza problem and some simple mass transport problems
3
Content available Ideal observability for bilinear discrete-time systems with and without delays in observation
88%
Lhous M. , Rachik M. , Magri E. M.
Archives of Control Sciences
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2018
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tom Vol. 28, No. 4
601--616
EN
An ideal observability subspace expression is stated for bilinear abstract system with bounded operator in Hilbert spaces. The case of finite dimentional space is also treated. However, it’s noticed that the state ideal observability can never be fulfilled within an infinite dimensional phase space in the case of scalar output. The case of bilinear discrete-time system with delays in observation is also described. To illustrate this work some examples are presented.
4
Content available Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality
88%
Ahmed N.
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nr 1
105-129
EN
In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions of optimality for partially observed relaxed controls. This is the main topic of this paper. Further we present an algorithm for computation of optimal policies followed by a brief discussion on regular versus relaxed controls. The paper is concluded by an example of a non-convex problem which is readily solvable by our approach.
5
Content available remote Commutativity up to a factor in Banach algebras
75%
Schmoeger C.
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tom Vol. 38, nr 4
895--900
EN
In this note we explore commutativity up to a factor ab = alfa ba for Hermitian or normal elements of a complex Banach algebra. Our results generalize results obtained for bounded linear operators on Hilbert spaces.
6
Content available Controllability of evolution equations and inclusions driven by vector measures
75%
Ahmed N. U.
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tom 24
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nr 1
49-72
EN
In this paper, we consider the question of controllability of a class of linear and semilinear evolution equations on Hilbert space with measures as controls. We present necessary and sufficient conditions for weak and exact (strong) controllability of a linear system. Using this result we prove that exact controllability of the linear system implies exact controllability of a perturbed semilinear system. Controllability problem for the semilinear system is formulated as a fixed point problem on the space of vector measures and is concluded controllability from the existence of a fixed point. Our results cover impulsive controls as well as regular controls.
7
Content available remote A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings
63%
Wega G. B. , Zegeye H. , Boikanyo O. A.
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2020
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tom Vol. 53, nr 1
152--166
EN
The purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.
8
Content available remote Dokładna rekonstrukcja stanu - teoria i przykłady zastosowania
63%
Byrski W.
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tom T. 7, z. 3
433-453
PL
Prezentowana jest ogólna teoria dokładnej rekonstrukcji skończenie wymiarowego wektora stanu w stacjonarnych układach liniowych, określonych w dowolnych przestrzeniach Hilberta. Dokładna rekonstrukcja stanu realizowana jest poprzez przetwarzanie sygnałów wejścia u i wyjścia y obserwowalnego systemu dynamicznego na zadanym przedziale czasowym T przez obserwator realizujący sumę iloczynu skalarnego pomiaru y i specjalnie dobranej funkcji obserwacji G1 oraz iloczynu skalarnego pomiaru u i specjalnie dobranej funkcji obserwacji G2 w wybranych przestrzeniach Hilberta. Funkcje G tworzące obserwator mają minimalną normę, co gwarantuje dodatkowe własności optymalności obserwatora w zadaniu minimalizacji błędu obserwacji stanu w obecności najbardziej niebezpiecznych znormalizowanych zakłóceń pomiarowych wejścia i wyjścia, przyjmowanych z kul jednostkowych.
EN
The general theory of exact reconstruction of finite dimensional state vector in time invariant linear systems is presented. The systems and signals are defined in Hilbert spaces. The exact reconstruction of the state is realized by processing of input u and output y signals in dynamic observable system on fix finite time interval T. The observer calculates the sum of inner product of y and special chosen observation function G1 and inner product of u and special observation function G2 in given Hilbert spaces. Observation functions G have minimal norm what guarantees the optimality properties of the observer in the task of minimization of observation error under noisy input/output measurement. Namely the most dangerous disturbances with bounded norm normalized to unit balls are assumed.
9
Content available Certain group dynamical systems induced by Hecke algebras
51%
Cho I.
Opuscula Mathematica
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2016
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tom Vol. 36, no. 3
337--373
EN
In this paper, we study dynamical systems induced by a certain group [formula] embedded in the Hecke algebra H(Gp) induced by the generalized linear group Gp = GL2(Qp) over the p-adic number fields Qp for a fixed prime p. We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms ol free probability on the Hecke algebra H(Gp).
10
Exact method of solution of torsional vibrations problem for discontinuous viscoelastic shaft with oscillators
51%
Cabański J.
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tom Vol. 42, no 4
459-474
EN
In this paper a generalized, exact method for solving the problems of torsional vibrations of a non-continuous viscoelastic shaft with oscillators has been presented. The oscillators are attached to the shaft by means of the viscoelastic constraints. The vibrations of this compound system are described by the set of conjugated, partial and ordinary differential equations. The separation of the variables in the complex Hilbert space, the obtained results from the analysis of the boundary-value problems and the proved generalized orthogonality of complex eigenvectors have been used to obtain the solution for free vibrations of the system at the arbitrary initial conditions. Finally, by applying the harmonic moment of the forced vibrations, the problem of the shaft has been solved. In the herein presented method, the operational generalized principles [6] have been applied.
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