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Stochastic viability and invariance. (English) Zbl 0741.60046

The paper extends Nagumo’s theorem on viability and/or invariance properties of closed subsets with respect to a differential equation [see, e.g., J. P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory (1984; Zbl 0538.34007)] to Itô’s differential equation \(d\xi=f(\xi)dt+g(\xi)dw(t)\), where \(f\) and \(g\) are Lipschitz functions defined on a closed convex subset \(K\). The main theorem involving \(K\), \(f\), \(g\) is deduced as a corollary of theorems on necessary conditions for viability and sufficient conditions for invariance in case when \(K\) is replaced by a set-valued random variable. The introduced main tools are the stochastic contingent and tangent sets to a set-valued random variable. The remaining sections are devoted to an elementary calculus of stochastic tangent sets to direct images, inverse images and intersections of closed subsets. An application to viable or controlled invariant stochastic linear control systems (\(f=A\xi+Bu\)) with \(K\) being a subspace of the state space can be regarded as extension of the one (\(g=0\)) in W. M. Wonham’s book [Linear multivariable control. A geometric approach (1985; Zbl 0609.93001)].
Reviewer: T.N.Pham (Hanoi)

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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References:

[1] J.-P. Aubin ( 1990 ), A survey of viability theory , SIAM J. Control & Optim . Zbl 0714.49021 · Zbl 0714.49021
[2] J.-P. Aubin (to appear), Viability Theory . · Zbl 0930.93051
[3] J.-P. Aubin - A. Cellina ( 1984 ), Differential Inclusions , Springer-Verlag . Zbl 0538.34007 · Zbl 0538.34007
[4] J.-P. Aubin - G. Da Prato ( 1990 ), Solutions contingentes de l’équation de la variété centrale , CRAS . 311 , p. 295 - 300 . Zbl 0714.34072 · Zbl 0714.34072
[5] J.-P. Aubin - G. Da Prato (to appear), Contingent Solutions to the Center Manifold Equation , Annales de l’Institut Henri Poincaré, Analyse Non Linéaire . Numdam | Zbl 0745.34039 · Zbl 0745.34039
[6] J.-P. Aubin - H. Frankowska , Set-Valued Analysis, Systems and Control: Foundations and Applications , Birkhäuser , Boston , Basel ( 1990 ). Zbl 0713.49021 · Zbl 0713.49021
[7] A. Bensoussan , Stochastic Control by Functional Analysis Methods , North-Holland ( 1982 ). Zbl 0474.93002 · Zbl 0474.93002
[8] A. Bensoussan - J.-L. Lions , Applications des inéquations variationnelles en contrôle stochastique , Dunod ( 1978 ). Zbl 0411.49002 · Zbl 0411.49002
[9] W.M. Wonham , Linear Multivariable Control. A Geometric Approach , Springer-Verlag ( 1985 ). Zbl 0609.93001 · Zbl 0609.93001
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