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2013, 2013(special): 407-414. doi: 10.3934/proc.2013.2013.407

Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion

1. 

Department of Mathematics and Mechanics, Saint-Petersburg State University, Saint-Petersburg, 198504, Russian Federation, Russian Federation

Received  September 2012 Published  November 2013

The present article consists of two parts. In the first part we consider evolutionary variational inequalities with a nonlinearity which is described by a differential inclusion. Using the frequency-domain method we prove, under certain assumptions, the dissipativity of our variational inequality which is important for the asymptotic behavior of the system. In the second part a coupled system of Maxwell's equation and the heat equation is considered. For this system we introduce the notion of stability on a finite-time interval and present a theorem on this type of stability.
Citation: Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407
References:
[1]

G. Duvant and J.L. Lions, "Inequalities in Mechanics and Physics,", Springer - Verlag, (1976).

[2]

D. Kalinichenko, V. Reitmann and S. Skopinov, Stability and bifurcations in a finite time interval on variational inequalities,, Differential Equations, 48 (2012), 1.

[3]

Y. Kalinin, V. Reitmann and N. Yumaguzin, Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect,, Discrete and Continuous Dynamical Systems - Supplement 2011, 2 (2011), 754.

[4]

A.L. Likhtarnikov and V.A. Yakubovich, The frequency theorem for equations of evolutionary type,, Siberian Math. J., 17 (1976), 790.

[5]

R.V. Manoranjan, H.M. Yin and R. Showalter, On two-phase Stefan problem arising from a microwave heating process,, Contin. and Discrete Dynamical Systems, 15 (2006), 1155.

[6]

A.N. Michel and D.W. Porter, Practical stability and finite-time stability of discontinuous systems,, IEEE Trans. Circuit Theory, 19 (1972), 123.

[7]

A.A. Pankov, "Bounded and Almost Periodic Solutions of Nonlinear Differential Operator Equations,", Naukova Dumka, (1986).

[8]

H. Triebel, "Interpolation Theorie, Function Spaces, Differential Operators,", Amsterdam, (1978).

[9]

L. Weiss and E.F. Infante, On the stability of systems defined over a finite time interval,, Proc. Nat. Acad. Sci., 54 (1965), 44.

show all references

References:
[1]

G. Duvant and J.L. Lions, "Inequalities in Mechanics and Physics,", Springer - Verlag, (1976).

[2]

D. Kalinichenko, V. Reitmann and S. Skopinov, Stability and bifurcations in a finite time interval on variational inequalities,, Differential Equations, 48 (2012), 1.

[3]

Y. Kalinin, V. Reitmann and N. Yumaguzin, Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect,, Discrete and Continuous Dynamical Systems - Supplement 2011, 2 (2011), 754.

[4]

A.L. Likhtarnikov and V.A. Yakubovich, The frequency theorem for equations of evolutionary type,, Siberian Math. J., 17 (1976), 790.

[5]

R.V. Manoranjan, H.M. Yin and R. Showalter, On two-phase Stefan problem arising from a microwave heating process,, Contin. and Discrete Dynamical Systems, 15 (2006), 1155.

[6]

A.N. Michel and D.W. Porter, Practical stability and finite-time stability of discontinuous systems,, IEEE Trans. Circuit Theory, 19 (1972), 123.

[7]

A.A. Pankov, "Bounded and Almost Periodic Solutions of Nonlinear Differential Operator Equations,", Naukova Dumka, (1986).

[8]

H. Triebel, "Interpolation Theorie, Function Spaces, Differential Operators,", Amsterdam, (1978).

[9]

L. Weiss and E.F. Infante, On the stability of systems defined over a finite time interval,, Proc. Nat. Acad. Sci., 54 (1965), 44.

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