Approximate Controllability of Semilinear Deterministic and Stochastic Evolution Equations in Abstract Spaces

Mahmudov, Nazım I.

doi:10.1137/s0363012901391688

Cited by 270 publications

(158 citation statements)

References 29 publications

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“…Roughly speaking, controllability generally means that it is possible to steer a dynamic control system from an arbitrary initial state to an arbitrary final state using a set of admissible controls. In the literature there are many different definitions of controllability, for both linear (Klamka, 1991;1993;Mahmudov, 2001a;Mahmudov and Denker, 2000) and nonlinear dynamic systems (Klamka, 2000;Mahmudov, 2002;2003b;Mahmudov and Zorlu, 2003), which strongly depend on the class of dynamic control systems and the set of admissible controls (Klamka, 1991;1996). Therefore, for deterministic linear and nonlinear dynamic systems there exist many different necessary and sufficient conditions for global and local controllability (Klamka, 1991;1993;1996;2000).…”

Section: Introductionmentioning

confidence: 99%

“…Stochastic controllability for finite dimensional nonlinear stochastic systems was discussed in (Arapostathis et al, 2001;Mohamed and Zorlu, 2003;Sunahara et al, 1974;1975). Using the theory of bounded nonlinear operators and linear semigroups, different types of stochastic controllability for nonlinear infinite dimensional control systems defined in Hilbert spaces were considered in (Mahmudov, 2002;2003b). In the papers (Klamka and Socha, 1974;1980), Lyapunov techniques were used to formulate and prove sufficient conditions for stochastic controllability of nonlinear finite dimensional stochastic systems with point delays in the state variable.…”

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confidence: 99%

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Stochastic Controllability of Systems with Multiple Delays in Control

Klamka¹

2009

International Journal of Applied Mathematics and Computer Science

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Finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with multiple point delays in control are considered. Using the notation, theorems and methods used for deterministic controllability problems for linear dynamic systems with delays in control as well as necessary and sufficient conditions for various kinds of stochastic relative controllability in a given time interval are formulated and proved. It will be proved that, under suitable assumptions, relative controllability of an associated deterministic linear dynamic system is equivalent to stochastic relative exact controllability and stochastic relative approximate controllability of the original linear stochastic dynamic system. As a special case, relative stochastic controllability of dynamic systems with a single point delay is also considered. Some remarks and comments on the existing results for stochastic controllability of linear dynamic systems are also presented.

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

Stochastic Controllability of Systems with Multiple Delays in Control

Klamka¹

2009

International Journal of Applied Mathematics and Computer Science

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“…Approximate controllability for fractional stochastic systems are well investigated, we refer to [10,11] and references therein.…”

Section: D Q T [Lx (T)] = (M + M) X (T) + Bu (T) + F (T X (T))mentioning

confidence: 99%

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

2017

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This paper investigates the approximate controllability for Sobolev type stochastic perturbed control systems of fractional order with fractional Brownian motion and Sobolev fractional stochastic nonlocal conditions in a Hilbert space, A new set of sufficient conditions are established by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is also given to illustrate the obtained theory.

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“…First formula is proved in [2]. The second one follows from the definition of the operator Π b s and the stochastic Fubini Theorem:…”

Section: U ) Introduce the Following Operatorsmentioning

confidence: 96%

Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

Mahmudov¹

2014

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.

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Approximate Controllability of Semilinear Deterministic and Stochastic Evolution Equations in Abstract Spaces

Cited by 270 publications

References 29 publications

Stochastic Controllability of Systems with Multiple Delays in Control

Stochastic Controllability of Systems with Multiple Delays in Control

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

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