Top

Published in:

2014 | OriginalPaper | Chapter

7. Dynamics and Control Under State Constraints

Authors : Alexander B. Kurzhanski, Pravin Varaiya

Published in: Dynamics and Control of Trajectory Tubes

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The topics of this chapter are problems of reachability and system dynamics under state constraints in the form of reach tubes. Indicated are general approaches based on the Hamiltonian formalism and a related Comparison Principle. Further emphasis is on the dynamics of linear systems under hard bounds on the controls and system trajectories. A detailed solution is presented based on ellipsoidal approximations of bounded trajectory tubes.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Impulse Controls and Double Constraints

next chapter Trajectory Tubes State-Constrained Feedback Control

The boundary \(\partial \mathcal{X}[\uptau ]\) of \(\mathcal{X}[\uptau ]\) may be defined as the set \(\partial \mathcal{X}[\uptau ] = \mathcal{X}[\uptau ]\setminus \mathrm{int}\mathcal{X}[\uptau ]\). Under the controllability assumption, \(\mathcal{X}[\uptau ]\) has a nonempty interior, \(\mathrm{int}\mathcal{X}[\uptau ]\not =\varnothing,\;\uptau> t_{0}\).

In the general case, under Assumption 7.2.1, the optimal trajectory may visit the smooth boundary of the state constraint for a countable set of closed intervals, and function L _∗ ⁰(⋅ ) allows not more than a countable set of discontinuities of the first order.

The same minimum value is also attained here in classes of functions broader than \(\mathcal{C}_{00}\).

The computer illustrations for this subsection were made by M.N. Kirilin.

Aubin, J.-P.: Viability Theory. SCFA. Birkhäuser, Boston (1991)MATH

Aubin, J.-P., Bayen, A.M., St.Pierre, P.: Viability Theory: New Directions. Springer, Heidelberg (2011)

16.

Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. SCFA. Birkhäuser, Boston (1997)CrossRefMATH

32.

Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston (2001)CrossRef

48.

Clarke, F.H., Ledyaev Yu, S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)

50.

Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–41 (1983)MathSciNetCrossRefMATH

51.

Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1 & 2. Wiley, New York (1989) (Wiley GmbH & Co., KGaA, 2008)

80.

Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)MATH

96.

Gusev, M.I., Kurzhanski, A.B.: Optimization of controlled systems with bounds on the controls and state coordinates, I. Differ. Equ. 7(9), 1591–1602 (1971) [translated from Russian]MATH

97.

Gusev, M.I., Kurzhanski, A.B.: Optimization of controlled systems with bounds on the controls and state coordinates, II. Differ. Equ. 7(10), 1789–1800 (1971) [translated from Russian]MATH

98.

Helton, J.W., James, M.R.: Extending H _∞ Control to Nonlinear Systems. SIAM, Philadelphia (1999)MATH

109.

Kalman, R.E., Ho, Y.-C., Narendra, K.S.: Controllability of linear dynamical systems. Contrib. Theory Differ. Equ. 1(2), 189–213 (1963)MathSciNet

123.

Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer Series in Soviet Mathematics. Springer, New York (1988)CrossRefMATH

137.

Kurzhanski, A.B.: Control and Observation Under Uncertainty Conditions, 392 pp. Nauka, Moscow (1977)

158.

Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes: a mathematical formalism for uncertain dynamics, viability and control. In: Advances in Nonlinear Dynamics and Control. Progress in Systems and Control Theory, vol. 17, pp. 122–188. Birkhäuser, Boston (1993)

160.

Kurzhanski, A.B., Kirilin, M.N.: Ellipsoidal techniques for reachability problems under nonellipsoidal constraints. In: Proceedings of the NOLCOS-01, St. Petersburg, pp. 768–768 (2001)

164.

Kurzhanski, A.B., Nikonov, O.I.: Evolution equations for trajectory bundles of synthesized control systems. Russ. Acad. Sci. Dokl. Math. 333(5), 578–581 (1993)

166.

Kurzhanski, A.B., Osipov, Y.S.: On optimal control under state constraints. Appl. Math. Mech. (PMM) 33(4), 705–719 (1969)

174.

Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhäuser, Boston (1997)CrossRefMATH

178.

Kurzhanski, A.B., Varaiya, P.: Dynamic optimization for reachability problems. J. Optim. Theory Appl. 108(2), 227–251 (2001)MathSciNetCrossRef

181.

Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. Part I: external approximations. Optim. Methods Softw. 17(2), 177–206 (2002)MathSciNetMATH

182.

Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. Part II: internal approximations. Box-valued constraints. Optim. Methods Softw. 17(2), 207–237 (2002)MathSciNetMATH

185.

Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability under state constraints. SIAM J. Control Optim. 45(4), 1369–1394 (2006)MathSciNetCrossRefMATH

198.

Leitmann, G.: Optimality and reachability with feedback controls. In: Blaquière, A., Leitmann, G. (eds.) Dynamical Systems and Microphysics: Control Theory and Mechanics. Academic, Orlando (1982)

221.

Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003)

234.

Riesz, F., Sz-.Nagy, B.: Leçons d’analyse fonctionnelle. Akadémiai Kiadó, Budapest (1972)

237.

Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University Press, Princeton (1999)

244.

Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)MATH

247.

Subbotin, A.I.: Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective. SCFA. Birkhäuser, Boston (1995)CrossRef

248.

Subbotina, N.N., Kolpakova, E.A., Tokmantsev, T.B., Shagalova, L.G.: Method of Characteristics for the Hamilton-Jacobi-Bellman Equation. Ural Scientific Center/Russian Academy of Sciences, Yekaterinburg (2013)

Title: Dynamics and Control Under State Constraints
Authors: Alexander B. Kurzhanski
Pravin Varaiya
Publisher: Springer International Publishing
Book: Dynamics and Control of Trajectory Tubes
Print ISBN: 978-3-319-10276-4

Electronic ISBN: 978-3-319-10277-1

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-10277-1_7

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner