Flow-invariant sets and differential inequalities in normed spaces<sup>†</sup>

“…The pair ( S, F ) is weakly invariant (respectively , strongly invariant) if for every [ t 0 , t 1 ) ∈Iand every x 0 ∈ S , at least one (respectively , every ) solution x ( · ) of (1) satisfies x ( t ) ∈ S for all t ∈ [ t 0 ,t 1 ). We refer the reader to [5,13,15] for the history of invariance theory , which is elsewhere called viability theory [1,2]. …”

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A Characterization of Strong Invariance for Perturbed Dissipative Systems

Donchev

Rios

Wolenski

2004

Optimal Control, Stabilization and Nonsmooth Analysis

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Summary. This note studies the strong invariance propertyo fadiscontinuous differential inclusion in whicht he righth and side is the sum of an almost upper semicontinuous dissipativea nd an almost lowers emicontinuous one-sided Lipschitz multifunction. Ac haracterization is obtained in terms of aHamilton-Jacobi inequality. 1I ntroduction and AssumptionsWe extend ac haracteriz ation of strong invariance that wasrecently provedin [16] to an onautonomous s y stem with weaker data assumptions. Consider the control s y stem modelled as ad ifferential inclusionwhere the given multifunction F : I×IR n → 2 IR n has compact convex values and I⊆IR is an open interval with [ t 0 ,t 1 ) ⊆I. The focus in this paper is on the case where the multifunction F has the formwhere D is upper semicontinuous and dissipative in x and G is lower semicontinuous and one-sided Lipschitz ; precise assumptions will be given below. We briefly recall the main concepts of invariant s y stems theory for the s y stem (1) using the language and notation from [6,7]. In addition to F , a closed set S ⊆ IR n is also given. The pair ( S, F ) is weakly invariant (respectively , strongly invariant) if for every [ t 0 , t 1 ) ∈Iand every x 0 ∈ S , at least one (respectively , every ) solution x ( · ) of (1) satisfies x ( t ) ∈ S for all t ∈ [ t 0 ,t 1 ). We refer the reader to [5,13,15] for the history of invariance theory , which is elsewhere called viability theory [1,2].

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“…To prove Theorem 1 and Theorem 2 we will use the following theorem (see [2] or [8]): Theorem 4. Let K 7 E be a normal cone with IntK j Y, and let f X 0Y T Â E 3 E be continuous and quasimonotone increasing.…”

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On ordinary differential equations with quasimonotone increasing right-hand side

Herzog

1998

Archiv der Mathematik

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We prove that the initial value problem x H t f tY xt, t P 0Y T, x0 x 0 , is uniquely solvable in certain ordered Banach spaces if, f is quasimonotone increasing with respect to x, and if f is satisfying a one±sided Lipschitz condition with respect to the order±inducing cone.1. Introduction. Let EY k Á k be a real Banach space. We consider a partial ordering % on E induced by a cone K with Int K j Y. A cone K is a closed convex subset of E with lK 7 K, l^0, and K ÀK f0g. Then x % y @A y À x P K, and we use the notation x ( y for y À x P Int K and K Ã for the dual cone, i.e., the set of all continuous linear functionals f on E with fx^0, x^0. The cone K is called normal if there is a g^1 with 0 % x % y A kxk % gkyk. For xY y P E, x % y we define the order interval xY y fz P E X x % z % yg. Now let f X 0Y TÂE 3 E be continuous, and let x 0 P E. We consider the initial value problem

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Flow-invariant sets and differential inequalities in normed spaces^†

Cited by 48 publications

References 8 publications

Smallest and greatest fixed points of quasimonotone increasing mappings

Smallest and greatest fixed points of quasimonotone increasing mappings

A Characterization of Strong Invariance for Perturbed Dissipative Systems

On ordinary differential equations with quasimonotone increasing right-hand side

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Flow-invariant sets and differential inequalities in normed spaces†

Cited by 48 publications

References 8 publications

Smallest and greatest fixed points of quasimonotone increasing mappings

Smallest and greatest fixed points of quasimonotone increasing mappings

A Characterization of Strong Invariance for Perturbed Dissipative Systems

On ordinary differential equations with quasimonotone increasing right-hand side

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Flow-invariant sets and differential inequalities in normed spaces^†