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  • 1
    Electronic Resource
    Electronic Resource
    Mutational equations in metric spaces (1993)
    Springer
    Set-valued analysis 1 (1993), S. 3-46 
    Details
    ISSN: 1572-932X
    Keywords: 34A99 ; 34G99 ; 49J99 ; 49N99 ; 49J52 ; 54G60 ; 46G05 ; Transitions ; mutations ; Nagumo ; center manifold ; Cauchy-Lipschitz ; Lyapunov method ; control ; visual ; mathematical morphology
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by ‘transitions’ with which we can also define differential quotients of a map. Their various limits are called ‘mutations’ of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations. This work was motivated by evolution equations of ‘tubes’ in ‘visual servoing’ on one hand, mathematical morphology on the other, when the metric spaces are ‘power spaces’. This paper begins by listing some consequences of general theorems concerning ‘mutational equations for tubes’.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01039289
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  • 2
    Electronic Resource
    Electronic Resource
    Introduction: Set-Valued Analysis in Control Theory (2000)
    Springer
    Set-valued analysis 8 (2000), S. 1-9 
    Details
    ISSN: 1572-932X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008724221942
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  • 3
    Electronic Resource
    Electronic Resource
    Stochastic Invariance for Differential Inclusions (2000)
    Springer
    Set-valued analysis 8 (2000), S. 181-201 
    Details
    ISSN: 1572-932X
    Keywords: stochastic invariance ; viability ; stochastic differential inclusions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this paper is to combine two ways for representing uncertainty through stochastic differential inclusions: a 'stochastic uncertainty", driven by a Wiener process, and a 'contingent uncertainty", driven by a set-valued map. The paper is also devoted to the invariance of closed under stochastic differential inclusions with a Lipschitz right-hand side, characterized in terms of stochastic tangent sets to closed subsets.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008738928302
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  • 4
    Electronic Resource
    Electronic Resource
    Stochastic Lyapunov method (1995)
    Springer
    Nonlinear differential equations and applications 2 (1995), S. 511-525 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is devoted to stability properties of solutions to stochastic differential equations obtained by a stochastic Lyapunov method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01210621
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  • 5
    Electronic Resource
    Electronic Resource
    Set-valued solutions to the Cauchy problem for hyperbolic systems of partial differential inclusions (1997)
    Springer
    Nonlinear differential equations and applications 4 (1997), S. 149-168 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions. The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations. As an application we construct the value function of the Mayer problem arising in control theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00001413
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  • 6
    Electronic Resource
    Electronic Resource
    Further properties of Lagrange multipliers in nonsmooth optimization (1980)
    Springer
    Applied mathematics & optimization 6 (1980), S. 79-90 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We compute the Clarke generalized gradient of the marginal function of a nonconvex optimization problem with respect to usual and non usual parameters and we show how Lagrange multipliers are involved in this formula.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01442884
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  • 7
    Electronic Resource
    Electronic Resource
    Boundary-value problems for systems of first-order partial differential inclusions (2000)
    Springer
    Nonlinear differential equations and applications 7 (2000), S. 67-90 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00001423
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  • 8
    Electronic Resource
    Electronic Resource
    In memory of Austin Blaquière (1923–1933) (1994)
    Springer
    Dynamics and control 4 (1994), S. 121-122 
    Details
    ISSN: 1573-8450
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01985937
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  • 9
    Electronic Resource
    Electronic Resource
    Monotone trajectories of multivalued dynamical systems (1977)
    Springer
    Annali di matematica pura ed applicata 115 (1977), S. 99-117 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We prove existence of « monotone trajectories » for a class of discrete and continuous systems sufficiently general to include problems of some interest in economic and biological theory. We prove existence of critical points which are Pareto minima. We study stability properties of Pareto minima.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02414712
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  • 10
    Electronic Resource
    Electronic Resource
    The Montagne Russe algorithm for global optimization (1998)
    Springer
    Mathematical methods of operations research 48 (1998), S. 153-168 
    Details
    ISSN: 1432-5217
    Keywords: Key words: Global optimization ; viability theory ; viability kernel ; Lyapunov function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract. The “Montagnes Russes” algorithm for finding the global minima of a lower semi-continuous function (thus involving state constraints) is a descent algorithm applied to an auxiliary function whose local and global minima are the global minima of the original function. Although this auxiliary function decreases along the trajectory of any of its minimizing sequences, the original function jumps above local maxima, leaves local minima, play “Montagnes Russes” (called “American Mountains” in Russian and “Big Dipper” in American!), but, ultimately, converges to its infimum. This auxiliary function is approximated by an increasing sequence of functions defined recursively at each point of the minimizing sequence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s001860050018
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