On Existence of Solutions to Di¡erential Equations or Inclusions Remaining in a Prescribed Closed Subset of a Finite-Dimensional Space (original) (raw)

Some results on existence and structure of solution sets to differential inclusions on the halfline

A topological structure of solution sets to multivalued differential problems on the halfline is studied by the use of Scorza-Dragoni type results and by the inverse systems approach. Some new existence results for asymptotic boundary value problems are also presented. Si studia la struttura topologica dell'insieme delle soluzioni di inclusioni differenziali sulla semiretta per mezzo di risultati tipo Scorza-Dragoni ed il metodo dei sistemi inversi. Sono anche presentati alcuni nuovi risultati di esistenza di soluzioni per problemi al bordo asintotici.

ON ATTAINABILITY OF A SET BY AT LEAST ONE SOLUTION TO A DIFFERENTIAL INCLUSION

In many dynamical system problems, it is interesting not only to know that equilibria do exist but also to know if the equilibria can be reached by at least one trajectory (possibly asymptotically). It is worth pointing out that this question is different from those of stability, or attractiveness of the equilibria. In the present article, our purpose is to give sufficient conditions for answering positively to the above question. More generally we address the question of attainability of a closed set E by at least one trajectory starting from outside E. Our method is based on a kind of Lyapunov Analysis based on Viability Theory and topological methods.

On the Existence of Solutions to Differential Inclusions with Nonconvex Right-Hand Sides

Siam Journal on Optimization - SIAMJO, 2007

We study the existence of solutions of differential inclusions with upper semicontinuous right-hand side. The investigation was prompted by the well known Filippov's examples. We define a new concept "colliding on a set". In the case when the admissible velocities do not "collide" on the set of discontinuities of the right-hand side, we expect that at least one trajectory emanates from every point. If the velocities do "collide" on the set of discontinuities of the right-hand side, the existence of solutions is not guaranteed, as is seen from one of the Filippov's examples. In this case we impose an additional condition in order to prove existence of a solution starting at a point of the discontinuity set. For the right-hand sides under consideration, we assume the following: whenever the velocities "collide" on a set S there exist tangent velocities (belonging to the Clarke tangent cone to S) on a dense subset of S. Then we prove existence of an ε-solution for every ε > 0. Under additional assumptions we can pass to the limit as ε → 0 and obtain a solution of the considered differential inclusion. Key words. differential inclusions with nonconvex right-hand side, existence of solutions; colliding on a set AMS subject classifications. 34A36, 34A60

Trajectories of differential inclusions with state constraints

Journal of Differential Equations, 2011

The paper deals with solutions of a differential inclusionẋ ∈ F (x) constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure coF satisfies a strict inward pointing condition at every boundary point x ∈ ∂Ω. Given a reference trajectory x * (·) taking values in an ε-neighborhood of Ω, we prove the existence of a second trajectory x : [0, T ] → Ω which satisfies x − x * W 1,1 ≤ Cε(1 + | ln ε|). As shown by an earlier counterexample, this bound is sharp.

Solutions of lower semicontinuous differential inclusions on closed sets

… del Seminario Matematico dell'Universit~ i di …, 1983

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