Abstract
We continue with the exponentiation analysis of multivalued maps defined on Banach spaces. In Part I of this work we have explored the Maclaurin exponentiation technique which is based on the use of a suitable power series. Now we focus the attention on the so-called recursive exponentiation method. Recursive exponentials are specially useful when it comes to study the reachable set associated to a differential inclusion of the form \(\dot z \in F(z)\). The definition of the recursive exponential of \(F: X \rightrightarrows X\) uses as ingredient the set of trajectories associated to the discrete time system \(z_{k+1}\in F(z_k)\). Although we are taking inspiration from a recent paper by Alvarez et al. [1] on the relation between continuous and discrete time evolution systems, our analysis and results go far beyond the particular context of convex processes considered by these authors.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alvarez, F., Correa, R., Gajardo, P.: Inner estimation of the eigenvalue set and exponential series solutions to differential inclusions. J. Convex Anal. 12, 1–11 (2005)
Amri, A., Seeger, A.: Exponentiating a bundle of linear operators. Set-Valued Anal. 14, (2006). DOI:10.1007/s11228-005-0007-z
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer, Dordrecht (1993)
Bogatyrev, A.V., Pyatnitskii, E.S.: Construction of piecewise-quadratic Lyapunov functions for nonlinear control systems (Russian). Avtomat i Telemekh. 10, 30–38 (1987)
Cabot, A., Seeger, A.: Multivalued exponentiation analysis. Part I. Maclaurin exponentials. Set-Valued Anal. (submitted)
Cominetti, R., Correa, R.: Sur une dérivée du second ordre en analyse non différentiable. C.R. Acad. Sci. Paris Sér I Math., 303(I), 861–864 (1986)
Kloeden, P.E., Valero, J.: Attractors of weakly asymptotically compact set-valued dynamical systems. Set-Valued Anal. 13, 381–404 (2005).
Hiriart-Urruty, J.B.: Characterization of the plenary hull of the generalized Jacobian matrix. Math. Programing 17, 1–12 (1982)
Ioffe, A.D.: Nonsmooth analysis: Differential calculus of nondifferentiable mappings. Trans. Amer. Math. Soc. 266, 1–56 (1981)
Molchanov, A.P., Pyatnitskii, E.S.: Lyapunov functions that determine necessary and sufficient conditions for the stability of linear differential inclusions (Russian). Nauka Sibirsk. Otdel. Novosibirsk 323, 52–61 (1987)
Molchanov, A.P., Pyatnitskiy, Y.S.: Stability criteria for selector-linear differential inclusions. Sovrem. Mat.—Stud. Aspir. 36, 421–424 (1988)
Molchanov, A.P., Pyatnitskiy, Y.S.: Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Systems Control. Lett. 13, 59–64 (1989)
Phat, V.N.: Weak asymptotic stabilizability of discrete-time systems given by set-valued operators. J. Math. Anal. Appl. 202, 363–378 (1996)
Phat, V.N.: Constrained Control Problems of Discret Processes. World Scientific, Singapore (1996)
Pyatnitskii, E.S., Rapoport, L.B.: Boundary of the domain of asymptotic stability of selector-linear differential inclusions and the existence of periodic solutions. Sovrem. Mat.—Stud. Aspir. 44, 785–790 (1992)
Pyatnitskii, E.S., Rapoport, L.B.: Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems. IEEE Trans. Circuits Systems I Fund. Theory Appl. 43, 219–229 (1996)
Rapaport, A., Sraidi, S., Terreaux, J.P.: Optimality of greedy and sustainable policies in the management of renewable resources. Optimal Control Appl. Methods 24, 23–44 (2003)
Rubinov, A.M., Makarov, V.L.: Mathematical Theory of Economic Dynamics and Equilibria. Springer, Berlin Heidelberg New York (1977)
Rubinov, A., Vladimirov, A.: Dynamics of positive multiconvex relations. J. Convex Anal. 8, 387–399 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cabot, A., Seeger, A. Multivalued Exponentiation Analysis. Set-Valued Anal 14, 381–411 (2006). https://doi.org/10.1007/s11228-006-0020-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-006-0020-x
Key words
- exponentiation
- multivalued map
- differential inclusion
- discrete trajectory
- reachable set
- Painlevé–Kuratowski limits