• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.213-230
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• 2002

We deal with the approximate controllability for semilinear systems with time delay in a Hilbert space. First, we show the existence and uniqueness of solutions of the given systems with the mere general Lipschitz continuity of nonlinear operator f from $R\;\times\;V$ to H. Thereafter, it is shown that the equivalence between the reachable set of the semilinear system and that of its corresponding linear system. Finally, we make a practical application of the conditions to the system with only discrete delay.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.81-97
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• 1999

This paper deals with the time optimal control problem for the retarded semilinear system by using the construction of fundamental solution in case where the principal operators are unbounded operators.

• Journal of the Korean Mathematical Society
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• v.36 no.2
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• pp.317-332
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• 1999

In this paper we deal with the optimal control problem for the semilinear functional differential equations with unbounded delays. We will also establish the regularity for solutions of the given system. By using the penalty function method we derive the optimal conditions for optimality of an admissible state-control pairs.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.469-481
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• 2020

In this paper, we consider a control system for semilinear differential equations in Hilbert spaces with Lipschitz continuous nonlinear term. Our method is to find the equivalence of approximate controllability for the given semilinear system and the linear system excluded the nonlinear term, which is based on results on regularity for the mild solution and estimates of the fundamental solution.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.479-497
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• 2018

This paper deals with an optimal control on semilinear stochastic functional differential equations with Poisson jumps in a Hilbert space. The existence of an optimal control is derived by the solution of proposed system which satisfies weakly sequentially compactness. Also the stochastic maximum principle for the optimal control is established by using spike variation technique of optimal control with a convex control domain in Hilbert space. Finally, an application of retarded type stochastic Burgers equation is given to illustrate the theory.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.355-368
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• 2021

In this paper, we investigate necessary and sufficient conditions for the approximate controllability for semilinear stochastic functional differential equations with delays in Hilbert spaces without the strict range condition on the controller even though the equations contain unbounded principal operators, delay terms and local Lipschitz continuity of the nonlinear term.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.361-376
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• 2000

Our purpose is to seek that the reachable set of the semilinear system $\frac{d}{dt}x(t){\;}={\;}Ax(t){\;}+{\;}f(t,x(t)){\;}+{\;}Bu(t)$ is equivalent to that of its corresponding to linear system (the case where f=0).Under the assumption that the system of generalized eigenspaces of A is complete, we will show that the reachable set corresponding to the linear system is independent of t in case A generates $C_0-semigroup$. An illustrative example for retarded system with time delay is given in the last section.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.603-611
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• 1996

Let consider the following problem: find an element u(t) in a Banach space U from the equation $$ x'(t) = Ax(t) + f(t,x(t)) + \Phi_0 u(t), 0 \leq t \leq T $$ with initial and terminal conditions $$ x(0) = 0, x(T) = \phi $$ in a Banach space X where $\phi \in D(A)$. This problem is a kind of control engineering inverse problem and contains nonlinear term, so that it is difficult and interesting. Thee proof main result in this paper is based on the Fredholm property of [1] in section 3. Similar considerations of linear system have been dealt with in many references. Among these literatures, Suzuki[5] introduced this problem for heat equation with unknown spatially-varing conductivity. Nakagiri and Yamamoto[2] considered the identifiability problem, which A is a unknown operator to be identified, where the system is described by a linear retarded functional differential equation. We can also apply to determining the magnitude of the control set for approximate controllability if X is a reflexive space, i.e., we can consider whether a dense subset of X is covered by reachable set in section 4.