Abstract
The problem of reachability for differential inclusions is an active topic in the recent control theory. Its solution provides an insight into the dynamics of an investigated system and also enables one to design synthesizing control strategies under a given optimality criterion. The primary results on reachability were mostly applicable to convex sets, whose dynamics is described through that of their support functions. Those results were further extended to the viability problem and some types of nonlinear systems. However, non-convex sets can arise even in simple bilinear systems. Hence, the issue of nonconvexity in reachability problems requires a more detailed investigation. The present article follows an alternative approach for this cause. It deals with star-shaped reachability sets, describing the evolution of these sets in terms of radial (Minkowski gauge) functions. The derived partial differential equation is then modified to cope with additional state constraints due to on-line measurement observations. Finally, the last section is on designing optimal closed-loop control strategies using radial functions.
Similar content being viewed by others
References
Panasyuk, A.I., Panasyuk, V.I.: An Equation Generated by a Differential Inclusion. Mathematical Notes of the Academy of Sciences of the USSR, 27:3, 213-218 (original Russian text published in Matematicheskiye Zametki, 27:3, 429437) (1980)
Panasyuk, A.I.: Differential equation for nonconvex attainment sets. Mathematical Notes of the Academy of Sciences of the USSR, 37:5, 395-400 (original Russian text published in Matematicheskiye Zametki, 37:5, 717–726) (1985)
Aubin, J.P., Cellina, A.: Differential Inclusions. Springer (1984)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. SCFA 2, Birkhauser, Boston (1990)
Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes - a mathematical formalism for uncertain dynamics, viability and control. Advances in Nonlinear Dynamics and Control, pp. 122–188. Birkhauser, Boston (1993)
Aubin, J.P., Bayen, A.M., St.Pierre, P.: Viability Theory: New Directions. Springer (2011)
Kurzhanski, A.B., Varaiya, P.: Dynamic Optimization for Reachability Problems. A Journal of Optimization Theory and Applications 108(2), 227–251 (2001)
Kurzhanski, A.B.: Selected Works of A.B. Kurzhanski. Moscow State University Pub., Moscow (partly in Russian) (2009)
Kurzhanski, A.B., Varaiya, P.: Dynamics and control of trajectory tubes. Birkhauser (2014)
Panasyuk, A.I.: Equation of Achievability as Applied to Optimal Control Problems. Automatization and Remote Control, Minsk, 43:5, 625636 (original Russian text published in Avtomatika and Telemekhanika (1982)
Aubin, J.P.: Viability Theory. SCFA. Birkhauser, Boston (1991)
Mazurenko, S.S.: A Differential Equation for the Gauge Function of the Star-Shaped Attainability Set of a Differential Inclusion. Doklady Mathematics, Moscow, 86:1, 476–479, (original Russian text published in Doklady Akademii Nauk, 445:2, 139142 (2012)
Kurzhanski, A.B., Varaiya, P.: A Comparison Principle for Equations of the Hamilton-Jacobi Type in Set-Membership Filtering. Commun. Inf. Syst. 6(3), 179–192 (2006)
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)
Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer (2004)
Fillippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Springer (1988)
Nadler, S.B.: Multi-Valued Contraction Mappings. Pacific J. Math. 30, 475–488 (1969)
Kirr, E., Petruel, A.: Continuous Dependence on Parameters of the Fixed Points Set for some Set-Valued Operators. Discussiones Mathematicae Differential Inclusions 17, 29–41 (1997)
Gel‘man, B.D.: Multivalued Contraction Maps and Their Applications, Vestnik Voronezhskogo Universiteta, Ser. Fiz. Mat., 1, 7486 (in Russian) (2009)
Crandall, M.G., Ishii, H., Lions, P-L: Users guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27.1, 1–67 (1992)
Sinyakov, V.: On external and internal approximations for reachability sets of bilinear systems. Doklady Mathematics 2(90) (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mazurenko, S. Partial Differential Equation for Evolution of Star-Shaped Reachability Domains of Differential Inclusions. Set-Valued Var. Anal 24, 333–354 (2016). https://doi.org/10.1007/s11228-015-0345-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-015-0345-4
Keywords
- Reachability sets
- Differential inclusion
- Star-shaped sets
- Radial (gauge) function
- Viability
- Optimal control synthesis