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1

Compactness in certain abstract function spaces with application to differential inclusions

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1995
|
tom 15
|
nr 1
75-94
EN
In this note we present a result on compactness in certain Banach spaces of vector valued functions. We demonstrate an application of this result to the questions of existence of solutions of nonlinear differential inclusions on a Banach space.
2

Compactness in certain abstract function spaces with application to differential inclusions

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1995
|
tom 15
|
nr 1
21-28
EN
In this note we present a result on compactness in certain Banach spaces of vector valued functions. We demonstrate an application of this result to the questions of existence of solutions of nonlinear differential inclusions on a Banach space.
3
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Impulsive perturbation of C₀-semigroups and stochastic evolution inclusions

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2002
|
tom 22
|
nr 1
125-149
EN
In this paper, we consider a class of infinite dimensional stochastic impulsive evolution inclusions. We prove existence of solutions and study properties of the solution set. It is also indicated how these results can be used in the study of control systems driven by vector measures.
4
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Optimal control of systems determined by strongly nonlinear operator valued measures

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2008
|
tom 28
|
nr 1
165-189
EN
In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V*. The system is given by dx + A(dt,x) = f(t,x)γ(dt) + B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0,T] where A is a strongly nonlinear operator valued measure mapping Σ × V to V* with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I × H to H, γ is a countably additive bounded positive measure and the control u is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems.
5
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Weak compactness in the space of operator valued measures $M_ba(Σ,𝓛(X,Y))$ and its applications

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2011
|
tom 31
|
nr 2
231-247
EN
In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{ba}(Σ,𝓛(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{ba}(Σ,𝓛₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.
6
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Measure valued solutions for stochastic evolution equations on Hilbert space and their feedback control

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2005
|
tom 25
|
nr 1
129-157
EN
In this paper, we consider a class of semilinear stochastic evolution equations on Hilbert space driven by a stochastic vector measure. The nonlinear terms are assumed to be merely continuous and bounded on bounded sets. We prove the existence of measure valued solutions generalizing some earlier results of the author. As a corollary, an existence result of a measure solution for a forward Kolmogorov equation with unbounded operator valued coefficients is obtained. The main result is further extended to cover Borel measurable drift and diffusion which are assumed to be bounded on bounded sets. Also we consider control problems for these systems and present several results on the existence of optimal feedback controls.
7
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Measure valued solutions for systems governed by neutral differential equations on Banach spaces and their optimal control

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2013
|
tom 33
|
nr 1
89-109
EN
In this paper we consider the question of existence of measure valued solutions for neutral differential equations on Banach spaces when there is no mild solutions. We prove the existence of measure solutions and their regularity properties. We consider also control problems of such systems and prove existence of optimal feedback controls for some interesting a-typical control problems.
8
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Vector and operator valued measures as controls for infinite dimensional systems: optimal control

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2008
|
tom 28
|
nr 1
95-131
EN
In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, we prove some results on existence of solutions and their regularity properties both for linear and semilinear systems. Also presented are results on continuous dependence of solutions on operator and vector valued measures, and other parameters determining the system which are then used to prove some results on control theory including existence and necessary conditions of optimality. Here the operator valued measures are treated as structural controls. The paper is concluded with some examples from classical and quantum mechanics and a remark on future direction.
9
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A relaxation theorem for partially observed stochastic control on Hilbert space

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2007
|
tom 27
|
nr 2
295-314
EN
In this paper, we present a result on relaxability of partially observed control problems for infinite dimensional stochastic systems in a Hilbert space. This is motivated by the fact that measure valued controls, also known as relaxed controls, are difficult to construct practically and so one must inquire if it is possible to approximate the solutions corresponding to measure valued controls by those corresponding to ordinary controls. Our main result is the relaxation theorem which states that the set of solutions corresponding to ordinary controls is weakly dense in the set of solutions corresponding to relaxed controls. This is presented in Theorem 5.3 after giving some existence results on optimal controls for the infinite dimensional Zakai equation used for its proof.
10
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Optimal control of ∞-dimensional stochastic systems via generalized solutions of HJB equations

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
|
tom 21
|
nr 1
97-126
EN
In this paper, we consider optimal feedback control for stochastc infinite dimensional systems. We present some new results on the solution of associated HJB equations in infinite dimensional Hilbert spaces. In the process, we have also developed some new mathematical tools involving distributions on Hilbert spaces which may have many other interesting applications in other fields. We conclude with an application to optimal stationary feedback control.
11
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Optimal control of impulsive stochastic evolution inclusions

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2002
|
tom 22
|
nr 2
155-184
EN
In this paper, we consider a class of infinite dimensional stochastic impulsive evolution inclusions driven by vector measures. We use stochastic vector measures as controls adapted to an increasing family of complete sigma algebras and prove the existence of optimal controls.
12
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Topological dual of $B_∞(I,𝓛 ₁(X,Y))$ with application to stochastic systems on Hilbert space

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2009
|
tom 29
|
nr 1
67-90
EN
In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to a stochastic structural control problem. An optimal operator-valued measure, considered as the structural control, is to be chosen so as to minimize fluctuation (volatility). Both existence of optimal policy and necessary conditions of optimality are presented including a conceptual algorithm.
13
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Infinite dimensional uncertain dynamic systems on Banach spaces and their optimal output feedback control

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2015
|
tom 35
|
nr 1
65-87
EN
In this paper we consider a class of partially observed semilinear dynamic systems on infinite dimensional Banach spaces subject to dynamic and measurement uncertainty. The problem is to find an output feedback control law, an operator valued function, that minimizes the maximum risk. We present a result on the existence of an optimal (output feedback) operator valued function in the presence of uncertainty in the system as well as measurement. We also consider uncertain stochastic systems and present similar results on the question of existence of optimal feedback laws.
14
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Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2014
|
tom 34
|
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.
15
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A general class of McKean-Vlasov stochastic evolution equations driven by Brownian motion and Lèvy process and controlled by Lèvy measure

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2016
|
tom 36
|
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
16
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Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2015
|
tom 35
|
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.
17

Measure solutions for semilinear evolution equations with polynomial growth and their optimal control

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1997
|
tom 17
|
nr 1-2
5-27
EN
In this paper we introduce a new concept of generalized solutions generalizing the notion of relaxed solutions recently introduced by Fattorini. We present some results on the question of existence of generalized or measure valued solutions for semilinear evolution equations on Banach spaces with polynomial nonlinearities. The results are illustrated by two examples one of which arises in nonlinear quantum mechanics. The results are then applied to some control problems.
18

Mathematical model and optimal control of flow induced vibration of pipelines

100%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1999
|
tom 19
|
nr 1-2
67-84
EN
In this paper we consider a dynamic model for flow induced vibration of pipelines. We study the questions of existence and uniqueness of solutions of the system. Considering the flow rate as the control variable, we present three different necessary conditions of optimality. The last one with state constraint involves Differential Inclusions. The paper is concluded with an algorithm for computing the optimal controls.
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