A note on the robust stability in the heat equation in a radially symmetric sphere
Abstract
In this work a robust stability concept is established for the heat equation in a radially symmetric sphere, using the concept of stability under constant action perturbations applied in ordinary differential equations. It is assumed that in the heat equation there is an external heat source that is represented via Fourier-Bessel series, and whose coefficients are bounded piecewise continuous functions. The method of separation of variables is applied to obtain solutions of the heat equation and the Fourier-Bessel coefficients are determined in such a way that the solution obtained, as well as its first partial derivatives, are bounded. Based on this, sufficient conditions are established to ensure robust stability in the heat equation.
Downloads
References
Aleksandrov, V. V., Reyes-Romero, M., Sidorenko, G. Y., and Temoltzi-Auila,R. (2010). Stability of controlled inverted pendulum under permanent horizontal perturbations of the supporting point. Mechanics of Solids,45(2):187–193
Bellman, R. (1948). Existence and boundedness of solutions of nonlinear partial differential equations of parabolic type. Transactions of the American Mathematical Society, 64(1):21–44.
Elsgoltz, L. (1983). Ecuaciones diferenciales y cálculo variacional. Mir, Moscú.
Hegyi, B. and Jung, S. M. (2013). On the stability of heat equation. Abstract and Applied Analysis. Article ID 202373, 4 pages.
Kostandian, B. A. (1960). On the stability of solutions of the nonlinear equation of heat conduction. Journal of Applied Mathematics and Mechanics, 24(6):1686–1689.
Kurzhanski, A. B. and Varaiya, P. (2014). Dynamics and control of trajectory tubes. Theory and computation. Systems & Control: Foundations & Applications. Springer, Switzerland.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1):3–21.
Tang, K. (2007). Fourier analysis, partial differential equations and variational methods. Number 3 in Mathematical Methods for Engineers and Scientists. Springer-Verlag, Berlin.
Temoltzi-Ávila, R. and Ávila-Pozos, R. (2020). Conjunto de alcanzabilidad de un sistema mecánico controlable y condiciones de estabilidad robusta. Revista Iberoamericana de Automática e Informática Industrial, 17(3):285–293.
Yang, X. (2019). Stability in measure for uncentain heat equation. Discrete and Continuous Dynamical Systems Series B, 24(12): 6533–6540.
Zhermolenko, V. N. and Temoltzi-Ávila, R. (2021). On the Bulgakov problem and the robust stability of hyperbolic equation. Moscow University Mechanics Bulletin, 76(4):95–104
