On the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems.

*(English)*Zbl 1250.93034Summary: A new systematic approach to the construction of approximate solutions to a class of nonlinear singularly perturbed feedback control systems using the boundary layer functions especially with regard to the possible occurrence of the boundary layers is proposed. For example, problems with feedback control, such as the steady-states of the thermostats, where the controllers add or remove heat, depending upon the temperature registered in another place of the heated bar, can be interpreted with a second-order ordinary differential equation subject to a nonlocal three-point boundary condition. The \(O(\epsilon )\) accurate approximation of behavior of these nonlinear systems in terms of the exponentially small boundary layer functions is given. At the end of this paper, we formulate the unsolved controllability problem for nonlinear systems.

##### MSC:

93B05 | Controllability |

93C10 | Nonlinear systems in control theory |

93C73 | Perturbations in control/observation systems |

93B52 | Feedback control |

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\textit{R. Vrabel}, Syst. Control Lett. 61, No. 3, 422--426 (2012; Zbl 1250.93034)

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##### References:

[1] | Munch, A.; Zuazua, E., Numerical approximation of null controls for the heat equation: ill-posedness and remedies, Inverse problems, 26, 8, (2010), Article Number: 085018 · Zbl 1203.35015 |

[2] | Castro, C.; Zuazua, E., Unique continuation and control for the heat equation from an oscillating lower dimensional manifold, SIAM journal on control and optimization, 43, 4, 1400-1434, (2004) · Zbl 1101.93010 |

[3] | Micu, S.; Zuazua, E., Regularity issues for the null-controllability of the linear 1-d heat equation, Systems & control letters, 60, 6, 406-413, (2011) · Zbl 1225.93027 |

[4] | Alvarez, J.J.; Gadella, M.; Glasser, L.M.; Lara, L.P.; Nieto, L.M., One dimensional systems with singular perturbations, Journal of physics: conference series, 284, 012009, (2011) |

[5] | Nguyen, T.; Gajic, Z., Finite horizon optimal control of singularly perturbed linear systems: a differential Lyapunov equation approach, IEEE transactions on automatic control, 55, 2148-2152, (2010) · Zbl 1368.93402 |

[6] | Kokotovic, P.; Khali, H.K.; O’Reilly, J., Singular perturbation methods in control, analysis and design, (1986), Academic Press London |

[7] | Artstein, Z., Stability in the presence of singular perturbations, Nonlinear analysis TMA, 34, 6, 817-827, (1998) · Zbl 0948.34029 |

[8] | Artstein, Z., Singularly perturbed ordinary differential equations with nonautonomous fast dynamics, Journal of dynamics and differential equations, 11, 297-318, (1999) · Zbl 0937.34044 |

[9] | Artstein, Z.; Gaitsgory, V., The value function of singularly perturbed control system, Applied mathematics and optimization, 41, 425-445, (2000) · Zbl 0958.49019 |

[10] | Gaitsgory, V., On a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems, SIAM journal on control and optimization, 43, 1, 325-340, (2004) · Zbl 1101.49023 |

[11] | Gaitsgory, V.; Nguyen, M.T., Multiscale singularly perturbed control systems: limit occupational measures sets and averaging, SIAM journal on control and optimization, 41, 3, 954-974, (2002) · Zbl 1027.34071 |

[12] | Bouzaouache, H.; Braiek, N.B., On guaranteed global exponential stability of polynomial singularly perturbed control systems, International journal of computers, communications & control, I, 4, 21-34, (2006) |

[13] | Bouzaouache, H.; Ennaceur, B.H.B.; Benrejeb, M., Reduced optimal control of nonlinear singularly perturbed systems, Systems analysis modelling simulation, 43, 1, 75-87, (2003) · Zbl 1057.93034 |

[14] | Christofides, P.D.; Teel, A.R., Singular perturbations and input-to-state stability, Institute of electrical and electronics engineers, transactions on automatic control, 41, 11, (1996) · Zbl 0864.93086 |

[15] | Mei, P.; Cai, Ch.; Zou, Y., A generalized KYP lemma-based approach for \(H_\infty\) control of singularly perturbed systems, Circuits, systems, and signal processing, (2009) |

[16] | Meng, B.; Jing, Y-w., Robust semiglobally practical stabilization for nonlinear singularly perturbed systems, Nonlinear analysis, 70, 2691-2699, (2009) · Zbl 1161.93022 |

[17] | I. Boglaev, Robust monotone iterates for nonlinear singularly perturbed boundary value problems, Boundary Value Problems Volume 2009, Article ID 320606, doi:10.1155/2009/320606. · Zbl 1177.65107 |

[18] | Jankowski, T., Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments, Applied mathematics and computation, 197, 179-189, (2008) · Zbl 1145.34355 |

[19] | Khan, R.A., Positive solutions of four-point singular boundary value problems, Applied mathematics and computation, 201, 762-773, (2008) · Zbl 1152.34016 |

[20] | Khan, R.A.; Webb, J.R.L., Existence of at least three solutions of a second-order three-point boundary value problem, Nonlinear analysis, 64, 1356-1366, (2006) · Zbl 1101.34005 |

[21] | Xu, X., Positive solutions for singular \(m\)-point boundary value problems with positive parameter, Journal of mathematical analysis and applications, 291, 352-367, (2004) · Zbl 1047.34016 |

[22] | Lin, X.; Liu, W., Singular perturbation of a second-order three-point boundary value problem for nonlinear systems, Journal of the Shanghai university (English edition), 13, 1, 16-19, (2009) · Zbl 1212.34027 |

[23] | Chang, K.W.; Howes, F.A., Nonlinear singular perturbation phenomena: theory and applications, (1984), Springer-Verlag New York · Zbl 0559.34013 |

[24] | De Coster, C.; Habets, P., () |

[25] | Vrabel, R., Nonlocal four-point boundary value problem for the singularly perturbed semilinear differential equations, Boundary value problems, 2011, 9, (2011), Article ID 570493 · Zbl 1209.34018 |

[26] | Miller, J.J.H.; ORiordan, E.; Shishkin, G.I., Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions, (1996), World Scientific Singapore |

[27] | O’ Riordan, E.; Quinn, J., Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems, BIT numerical mathematics, 51, 317-337, (2011) · Zbl 1229.65130 |

[28] | Shishkin, G.I., Grid approximation of singularly perturbed parabolic convection – diffusion equations subject to a piecewise smooth initial condition, Computational mathematics and mathematical physics, 46, 1, 49-72, (2006) · Zbl 1210.35148 |

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