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Search Results (5)

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Keywords = semilinear heat equation

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16 pages, 329 KiB  
Article
Finite-Approximate Controllability of ν-Caputo Fractional Systems
by Muath Awadalla, Nazim I. Mahmudov and Jihan Alahmadi
Fractal Fract. 2024, 8(1), 21; https://doi.org/10.3390/fractalfract8010021 - 26 Dec 2023
Cited by 2 | Viewed by 1516
Abstract
This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control [...] Read more.
This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear ν-Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear ν-Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to ν-Caputo fractional heat equations. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
20 pages, 339 KiB  
Article
Mean Square Finite-Approximate Controllability of Semilinear Stochastic Differential Equations with Non-Lipschitz Coefficients
by Nazim I. Mahmudov
Mathematics 2023, 11(3), 639; https://doi.org/10.3390/math11030639 - 27 Jan 2023
Cited by 3 | Viewed by 1495
Abstract
In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean [...] Read more.
In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean square finite-approximate controllability of linear stochastic evolution systems. A control is also found that provides finite-dimensional mean exact controllability in addition to the requirement of approximate mean square controllability. Under the assumption of approximate mean square controllability of the associated linear stochastic system, we obtain sufficient conditions for the mean square finite-approximate controllability of a semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients using the Picard-type iterations. An application to stochastic heat conduction equations is considered. Full article
24 pages, 386 KiB  
Article
Finite-Approximate Controllability of Riemann–Liouville Fractional Evolution Systems via Resolvent-Like Operators
by Nazim I. Mahmudov
Fractal Fract. 2021, 5(4), 199; https://doi.org/10.3390/fractalfract5040199 - 4 Nov 2021
Cited by 9 | Viewed by 2102
Abstract
This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like [...] Read more.
This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like operators. We also find that such a control provides finite-dimensional exact controllability in addition to the approximate controllability requirement. Assuming the finite-approximate controllability of the corresponding linearized RL fractional evolution equation, we obtain sufficient conditions for finite-approximate controllability of the semilinear RL fractional evolution equation under natural conditions. The results are a generalization and continuation of recent results on this subject. Applications to fractional heat equations are considered. Full article
(This article belongs to the Special Issue Qualitative Analysis of Fractional Deterministic and Stochastic Systems)
15 pages, 356 KiB  
Article
On a Semilinear Parabolic Problem with Four-Point Boundary Conditions
by Marián Slodička
Mathematics 2021, 9(5), 468; https://doi.org/10.3390/math9050468 - 25 Feb 2021
Cited by 2 | Viewed by 1579
Abstract
This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability [...] Read more.
This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability of a steady-state problem is addressed and a constructive algorithm for finding a solution is proposed. Combining this schema with the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem is shown using the semigroup theory in C-spaces. Numerical experiments support the theoretical algorithms. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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16 pages, 311 KiB  
Article
Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations
by Ratinan Boonklurb, Tawikan Treeyaprasert and Aong-art Wanna
Symmetry 2020, 12(12), 2075; https://doi.org/10.3390/sym12122075 - 14 Dec 2020
Viewed by 1846
Abstract
This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for [...] Read more.
This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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