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**A survey of viability theory.**
*(English)*
Zbl 0714.49021

Summary: Some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state-dependent control constraints are motivated and surveyed. They all deal with viable solutions to nonlinear control problems, i.e., solutions satisfying at each instant given state constraints of a general and diverse nature. Some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case, including inequality constraints bearing on the state and state-dependent constraints on the controls. For instance, existence of a viability kernel of a closed set (corresponding to the largest controlled invariant manifold) is provided under general conditions, even when the zero-dynamics algorithm does not converge.

The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks. Viability theorems also allow the extension of Lyapunov’s second method to nonsmooth observation functions and the construction of “best” Lyapunov functions. As an application, “fuzzy differential inclusion” is presented.

Proofs and complements can be found in the author’s work, “Viability theory” (1991, to appear). They rely on properties of differential inclusion [see the author and A. Cellina, “Differential inclusions. Set-valued maps and viability theory” (1984; Zbl 0538.34007)] and set-valued analysis, [see the author, “Set-valued analysis” (Basel 1990)].

The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks. Viability theorems also allow the extension of Lyapunov’s second method to nonsmooth observation functions and the construction of “best” Lyapunov functions. As an application, “fuzzy differential inclusion” is presented.

Proofs and complements can be found in the author’s work, “Viability theory” (1991, to appear). They rely on properties of differential inclusion [see the author and A. Cellina, “Differential inclusions. Set-valued maps and viability theory” (1984; Zbl 0538.34007)] and set-valued analysis, [see the author, “Set-valued analysis” (Basel 1990)].

### MSC:

49J52 | Nonsmooth analysis |

93C10 | Nonlinear systems in control theory |

26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |

26A51 | Convexity of real functions in one variable, generalizations |

26E25 | Set-valued functions |

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

28D05 | Measure-preserving transformations |

34A60 | Ordinary differential inclusions |

34Dxx | Stability theory for ordinary differential equations |

39Axx | Difference equations |

54C60 | Set-valued maps in general topology |

54C65 | Selections in general topology |

58C06 | Set-valued and function-space-valued mappings on manifolds |

58C07 | Continuity properties of mappings on manifolds |

58C30 | Fixed-point theorems on manifolds |

93C15 | Control/observation systems governed by ordinary differential equations |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |