Partial differential equation for evolution of star-shaped reachability domains of differential inclusions.

*(English)*Zbl 1338.93065Summary: The problem of reachability for differential inclusions is an active topic in the recent control theory. Its solution provides an insight into the dynamics of an investigated system and also enables one to design synthesizing control strategies under a given optimality criterion. The primary results on reachability were mostly applicable to convex sets, whose dynamics is described through that of their support functions. Those results were further extended to the viability problem and some types of nonlinear systems. However, non-convex sets can arise even in simple bilinear systems. Hence, the issue of nonconvexity in reachability problems requires a more detailed investigation. The present article follows an alternative approach for this cause. It deals with star-shaped reachability sets, describing the evolution of these sets in terms of radial (Minkowski gauge) functions. The derived partial differential equation is then modified to cope with additional state constraints due to on-line
measurement observations. Finally, the last section is on designing optimal closed-loop control strategies using radial functions.

##### MSC:

93B03 | Attainable sets, reachability |

49K15 | Optimality conditions for problems involving ordinary differential equations |

##### Keywords:

reachability sets; differential inclusion; star-shaped sets; radial (gauge) function; viability; optimal control synthesis
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\textit{S. Mazurenko}, Set-Valued Var. Anal. 24, No. 2, 333--354 (2016; Zbl 1338.93065)

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##### References:

[1] | Panasyuk, A.I., Panasyuk, V.I.: An Equation Generated by a Differential Inclusion. Mathematical Notes of the Academy of Sciences of the USSR, 27:3, 213-218 (original Russian text published in Matematicheskiye Zametki, 27:3, 429437) (1980) · Zbl 0466.49032 |

[2] | Panasyuk, A.I.: Differential equation for nonconvex attainment sets. Mathematical Notes of the Academy of Sciences of the USSR, 37:5, 395-400 (original Russian text published in Matematicheskiye Zametki, 37:5, 717-726) (1985) · Zbl 0601.49026 |

[3] | Aubin, J.P., Cellina, A.: Differential Inclusions. Springer (1984) · Zbl 0538.34007 |

[4] | Aubin, J.P., Frankowska, H.: Set-Valued Analysis. SCFA 2, Birkhauser, Boston (1990) · Zbl 0713.49021 |

[5] | Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes - a mathematical formalism for uncertain dynamics, viability and control. Advances in Nonlinear Dynamics and Control, pp. 122-188. Birkhauser, Boston (1993) · Zbl 0912.93040 |

[6] | Aubin, J.P., Bayen, A.M., St.Pierre, P.: Viability Theory: New Directions. Springer (2011) · Zbl 1238.93001 |

[7] | Kurzhanski, AB; Varaiya, P, Dynamic optimization for reachability problems, A Journal of Optimization Theory and Applications, 108, 227-251, (2001) · Zbl 1033.93005 |

[8] | Kurzhanski, A.B.: Selected Works of A.B. Kurzhanski. Moscow State University Pub., Moscow (partly in Russian) (2009) · Zbl 1181.93010 |

[9] | Kurzhanski, A.B., Varaiya, P.: Dynamics and control of trajectory tubes. Birkhauser (2014) · Zbl 1336.93004 |

[10] | Panasyuk, A.I.: Equation of Achievability as Applied to Optimal Control Problems. Automatization and Remote Control, Minsk, 43:5, 625636 (original Russian text published in Avtomatika and Telemekhanika (1982) |

[11] | Aubin, J.P.: Viability Theory. SCFA. Birkhauser, Boston (1991) |

[12] | Mazurenko, S.S.: A Differential Equation for the Gauge Function of the Star-Shaped Attainability Set of a Differential Inclusion. Doklady Mathematics, Moscow, 86:1, 476-479, (original Russian text published in Doklady Akademii Nauk, 445:2, 139142 (2012) · Zbl 1357.93008 |

[13] | Kurzhanski, AB; Varaiya, P, A comparison principle for equations of the Hamilton-Jacobi type in set-membership filtering, Commun. Inf. Syst., 6, 179-192, (2006) · Zbl 1132.93024 |

[14] | Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970) · Zbl 0193.18401 |

[15] | Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer (2004) |

[16] | Fillippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Springer (1988) |

[17] | Nadler, SB, Multi-valued contraction mappings, Pacific J. Math., 30, 475-488, (1969) · Zbl 0187.45002 |

[18] | Kirr, E; Petruel, A, Continuous dependence on parameters of the fixed points set for some set-valued operators, Discussiones Mathematicae Differential Inclusions, 17, 29-41, (1997) · Zbl 0904.47042 |

[19] | Gel‘man, B.D.: Multivalued Contraction Maps and Their Applications, Vestnik Voronezhskogo Universiteta, Ser. Fiz. Mat., 1, 7486 (in Russian) (2009) · Zbl 0187.45002 |

[20] | Crandall, M.G., Ishii, H., Lions, P-L: Users guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27.1, 1-67 (1992) · Zbl 0755.35015 |

[21] | Sinyakov, V.: On external and internal approximations for reachability sets of bilinear systems. Doklady Mathematics 2(90) (2014) · Zbl 1304.93026 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.