American Institute of Mathematical Sciences

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2015, 2015(special): 579-587. doi: 10.3934/proc.2015.0579

On reachability analysis for nonlinear control systems with state constraints

Mikhail Gusev 1,

1. 

N.N.Krasovskii Institute of Mathematics and Mechanics, S.Kovalevskaya str., 16, 620099, Ekaterinburg, Russian Federation

Received  September 2014 Revised  February 2015 Published  November 2015

The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set for a nonlinear inequality. A procedure to remove state constraints is proposed; this procedure consists in replacing a primary system by an auxiliary system without state constraints. The equations of the auxiliary system depend on a small parameter. It is shown that a reachable set of the primary system may be approximated in the Hausdorff metric by reachable sets of the auxiliary system when the small parameter tends to zero. The estimates of the rate of convergence are given.
Keywords: approximation, penalty function, Hausdor metric., Reachable set, state constraints.
Mathematics Subject Classification: Primary: 93B03, 93C15; Secondary: 49J1.
Citation: Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579
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show all references

References:
[1]

SIAM J. Optim., 18 (2007), 1004-1026.  Google Scholar

[2]

SIAM J. Control Optim., 48 (2010), 4664-4679.  Google Scholar

[3]

Applied Mathematics, 3 (2012), 1593-1597. Google Scholar

[4]

J.Differential Integral Equations, 12 (1999), 471-497.  Google Scholar

[5]

J. Optim. Theory Appl., 104 (2000), 21-40.  Google Scholar

[6]

Sbornik Mathematics, 196 (2005), 513-539. Google Scholar

[7]

Proceedings of the Steklov Institute of Mathematics, 275, Suppl.1 (2011), 57-67. Google Scholar

[8]

Automation and Remote Control, 73 (2012), 450-461.  Google Scholar

[9]

Proceedings of the Steklov Institute of Mathematics 287 (2014), 77-92. Google Scholar

[10]

Studies in Mathematics and its Applications, Amsterdam : North-Holland, 1979.  Google Scholar

[11]

Automation and Remote Control, 72 (2011), 1841-1851.  Google Scholar

[12]

Differentsial'nye Uravneniya, 23 (1987), 1303-1315.  Google Scholar

[13]

J. Optim. Theory Appl., 128 (2006), 499-521.  Google Scholar

[14]

SCFA. Boston: Birkhäuser, 1997.  Google Scholar

[15]

New York: Wiley, 1967.  Google Scholar

[16]

GAMM Mitt. Ges. Angew. Math. Mech., 21 (1998), 103-135  Google Scholar

[17]

Z. Vycisl. Mat. i Mat. Fiz, 15 (1975), 67-78.  Google Scholar

[18]

System and Control Letters, 13 (1989), 117-123.  Google Scholar

[19]

SIAM J. Control and Optim. 43 (2004), 697-707.  Google Scholar

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