Abstract
The properties of invariance, stability, asymptotic stability and attainability of a given compact set \(K \subset \mathbb{R}^n \) with respect to a differential inclusion, have weak and strong versions: the weak version requires existence of a trajectory with the corresponding property, while the strong one requires this property for all trajectories. The following statement is proven in the paper (under slight restrictions) for each of the above-mentioned properties: if K has the weak property with respect to \(\dot x \in F(x) \), then there is a (regulation) mapping G such that G(x) ⊂ F(x) ∀ x and G has the strong property with respect to \({\dot x}\) ε G(x). In addition, certain regularity of the set of solutions of the last inclusion is claimed.
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References
Artstein, Z.: Stabilization with relaxed controls, Nonlinear Anal. 7 (1983), 1163–1173.
Aubin, J.-P. and Cellina, A.: Differential Inclusions, Springer-Verlag, Berlin, 1984.
Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin, 1990.
Cannarsa, P. and Sinestrari, C.: Convexity properties of the minimal time function, Calculus of Variations, Preprint.
Clarke, F., Ledyaev, Y., Stern, R. and Wolenski, P.: Qualitative properties of trajectories of control systems: a survey, J. Dynamic Control Systems 1(1) (1995), 1–48.
Clarke, F. and Wolenski, P.: Control systems to sets and their interiors, J. Optim. Theory Appl. 88(1) (1996), 3–23.
Deimling, K.: Multivalued Differential Equations, Walter de Gruyter, Berlin, New York, 1992.
Kurzhanski, A. and Filippova, T.: On the theory of trajectory tubes – a mathematical formalism for uncertain dynamics, viability and control, in Advances in Nonlinear Dynamics and Control: a Report from Russia, Progr. Systems Control Theory 17, Birkhäuser, Boston, 1993, pp. 122–188.
Ledyaev, Y.: Criteria for Viability of Trajectories of Nonautonomous Differential Inclusion and Their Applications, Technical Report CRM 1583, Université de Montréal, Montéal, Canada, 1989.
Taubert, K.: Converging multistep methods for initial value problems involving multivalued maps, Computing 27 (1981), 123–136.
Veliov, V.: On the Lipschitz continuity of the value function in optimal control, Preprint, Jan. 1993, to appear in J. Optim. Theory Appl.
Veliov, V.: Guaranteed Control of Uncertain Systems: Funnel Equations and Existence of Regulation Maps, Working Paper WP-92-62, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1992.
Veliov, V.: Sufficient conditions for viability under imperfect measurement, Set-Valued Anal. 1 (1993), 305–317.
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Veliov, V. Stability-Like Properties of Differential Inclusions. Set-Valued Analysis 5, 73–88 (1997). https://doi.org/10.1023/A:1008683223676
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DOI: https://doi.org/10.1023/A:1008683223676