Abstract
In this chapter we consider singularly perturbed differential systems whose degenerate equations have an isolated but not simple solution. In that case, the standard theory to establish a slow integral manifold near this solution does not work. Applying scaling transformations and using the technique of gauge functions we reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate the method by several examples from control theory and chemical kinetics.
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Notes
- 1.
A stable matrix is one whose eigenvalues all have strictly negative real parts.
References
Gu, Z., Nefedov, N.N., O’Malley, R.E.: On singular singularly perturbed initial values problems. SIAM J. Appl. Math. 49, 1–25 (1989)
Kalachev, L.V., O’Malley, R.E.: The regularization of linear differential-algebraic equations. SIAM J. Math. Anal. 27(1), 258–273 (1996)
Kokotović, P.V., Khalil, K.H., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. SIAM, Philadelphia (1986)
Lam, S.H., Goussis, D.M.: The GSP method for simplifying kinetics. Int. J. Chem. Kinet. 26, 461–486 (1994)
Lancaster, P.: Theory of Matrices. Academic, New York/London (1969)
Matyukhin, V.I.: Stability of the manifolds of controlled motions of a manipulator. Autom. Remote Control 59(4), 494–501 (1998)
O’Malley, R.E.: High gain feedback systems as singular singular–perturbation problems. In: Prepr. techn. pap., 18th Joint Automat. Control Conf., New York, pp. 1278–1281 (1977)
Sobolev, V.A.: Geometrical theory of singularly perturbed control systems. In: Proceedings of the 11th Congress of IFAC, Tallinn, vol. 6, pp. 163–168 (1990)
Sobolev, V.A., Tropkina, E.A.: Asymptotic expansions of slow invariant manifolds and reduction of chemical kinetics models. Comput. Math. Math. Phys. 52, 75–89 (2012)
Stewart, G.W.: Introduction to Matrix Computations. Academic, New York (1973)
Utkin, V.I.: Application of equivalent control method to the systems with large feedback gain. IEEE Trans. Automat. Control 23(3), 484–486 (1977)
Utkin, V.I.: Sliding Modes and Their Applications in Variable Structure Systems. Mir, Moscow (1978)
Vasil’eva, A.B., Butuzov, V.F., Kalachev, L.V.: The boundary function method for singular perturbation problems. In: Studies in Applied Mathematics, vol. 14. SIAM, Philadelphia (1995)
Young, K.-K.D., Kokotović, P.V., Utkin, V.I.: A singular perturbation analysis of high-gain feedback systems. IEEE Trans. Automat. Control 22(6), 931–938 (1977)
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Shchepakina, E., Sobolev, V., Mortell, M.P. (2014). Singular Singularly Perturbed Systems. In: Singular Perturbations. Lecture Notes in Mathematics, vol 2114. Springer, Cham. https://doi.org/10.1007/978-3-319-09570-7_5
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DOI: https://doi.org/10.1007/978-3-319-09570-7_5
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