Abstract
We examine linear, time invariant, singularly perturbed differential equations, where a split into slow and fast variables is not prescribed. Simple linear algebra considerations give rise to a useful order reduction type framework. A comparison with the classical order reduction method is provided, and the relation to efficient computations is pointed out.
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Artstein, Z.: On singularly perturbed ordinary differential equations with measure-valued limits. Math. Bohem. 127, 139–152 (2002)
Artstein, Z., Gear, C.W., Kevrekidis, I.G., Slemrod, M., Titi, E.S.: Analysis and computation of a discrete KdV-Burgers type equation with fast dispersions and slow diffusion. SIAM J. Numer. Anal. 49, 2124–2143 (2011)
Artstein, Z., Kevrekidis, I.G., Slemrod, M., Titi, E.S.: Slow observables of singularly perturbed differential equations. Nonlinearity 20, 2463–2481 (2007)
Chatterjee, S., Acharya, A., Artstein, Z.: Computing singularly perturbed differential equations. J. Comput. Phys. 354, 417–446 (2018)
Ash, R.B.: Real Analysis and Probability. Academic Press, New York (1972)
Donchev, A.L.: Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. Lecture Notes in Control and Information Science, vol. 52. Springer, Berlin (1983)
Dontchev, A.L., Veliov, V.M.: Singular perturbation in Mayer’s problem for linear systems. SIAM J. Control. Optim. 21, 566–581 (1983)
Dontchev, A.L., Veliov, V.M.: Singular perturbations in linear control systems with weakly coupled stable and unstable fast subsystems. J. Math. Anal. Appl. 110, 1–30 (1985)
Dontchev, A.L., Veliov, V.M.: On the order reduction of linear optimal control systems in critical cases. In: Bagchi, A., Jongen, H.T. (Eds.) Systems and Optimization, Lecture Notes in Control and Inform. Sci., 66, pp. 61–73. Springer, Berlin (1985)
Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York (1974)
Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Methods, Singular Perturbation, in Control: Analysis and Design. Academic Press, London,: reprinted as Classics in Applied Mathematics 25, p. 1999. SIAM Publications, Philadelphia (1986)
Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8, 1269–1324 (2010)
Tikhonov, A.N., Vasiléva, A.B., Sveshnikov, A.G.: Differential Equations. Springer, Berlin (1985)
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In memory of a great scholar, Asen L. Dontchev.
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Artstein, Z. Linear singularly perturbed systems without slow-fast split. Comput Optim Appl 86, 871–884 (2023). https://doi.org/10.1007/s10589-022-00412-9
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DOI: https://doi.org/10.1007/s10589-022-00412-9