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
References:
[1]

R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler method for state-constrained differential inclusions, SIAM J. Optim., 18 (2007), 1004-1026.  Google Scholar

[2]

P. Bettiol, A. Bressan, R. Vinter, Trajectories Satisfying a State Constraint: $W^{(1,1)}$ Estimates and Counterexamples, SIAM J. Control Optim., 48 (2010), 4664-4679.  Google Scholar

[3]

N. Bonneuil, Computing reachable sets as capture-viability kernels in reverse time, Applied Mathematics, 3 (2012), 1593-1597. Google Scholar

[4]

F. Forcellini and F. Rampazzo, On non-convex differential inclusions whose state is constrained in the closure of an open set, J.Differential Integral Equations, 12 (1999), 471-497.  Google Scholar

[5]

H. Frankowska and R. B. Vinter, Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 21-40.  Google Scholar

[6]

S. V. Grigor'eva, V. Y. Pakhotinskikh, A. A. Uspenskii and V. N. Ushakov, Construction of solutions in certain differential games with phase constraints, Sbornik Mathematics, 196 (2005), 513-539. Google Scholar

[7]

M. I. Gusev, On external estimates for reachable sets of nonlinear control systems, Proceedings of the Steklov Institute of Mathematics, 275, Suppl.1 (2011), 57-67. Google Scholar

[8]

M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Automation and Remote Control, 73 (2012), 450-461.  Google Scholar

[9]

M. I. Gusev, Internal approximations of reachable sets of control systems with state constraints, Proceedings of the Steklov Institute of Mathematics 287 (2014), 77-92. Google Scholar

[10]

A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems", Studies in Mathematics and its Applications, Amsterdam : North-Holland, 1979.  Google Scholar

[11]

E. K. Kostousova, On polyhedral estimates for reachable sets of multistep systems with bilinear uncertainty, Automation and Remote Control, 72 (2011), 1841-1851.  Google Scholar

[12]

A. B. Kurzhanski and T. F. Filippova, Description of the pencil of viable trajectories of a control system(Russian), Differentsial'nye Uravneniya, 23 (1987), 1303-1315.  Google Scholar

[13]

A. B. Kurzhanski, I. M. Mitchell and P. Varaiya, Optimization techniques for state-constrained control and obstacle problems, J. Optim. Theory Appl., 128 (2006), 499-521.  Google Scholar

[14]

A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control", SCFA. Boston: Birkhäuser, 1997.  Google Scholar

[15]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory", New York: Wiley, 1967.  Google Scholar

[16]

F. Lempio and V. M. Veliov, Discrete approximations of differential inclusions, GAMM Mitt. Ges. Angew. Math. Mech., 21 (1998), 103-135  Google Scholar

[17]

A.V. Lotov, A numerical method for constructing sets of attainability for linear controlled systems with phase constraints (Russian), Z. Vycisl. Mat. i Mat. Fiz, 15 (1975), 67-78.  Google Scholar

[18]

E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, System and Control Letters, 13 (1989), 117-123.  Google Scholar

[19]

R. J. Stern, Characterization of the State Constrained Minimal Time Function, SIAM J. Control and Optim. 43 (2004), 697-707.  Google Scholar

show all references

References:
[1]

R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler method for state-constrained differential inclusions, SIAM J. Optim., 18 (2007), 1004-1026.  Google Scholar

[2]

P. Bettiol, A. Bressan, R. Vinter, Trajectories Satisfying a State Constraint: $W^{(1,1)}$ Estimates and Counterexamples, SIAM J. Control Optim., 48 (2010), 4664-4679.  Google Scholar

[3]

N. Bonneuil, Computing reachable sets as capture-viability kernels in reverse time, Applied Mathematics, 3 (2012), 1593-1597. Google Scholar

[4]

F. Forcellini and F. Rampazzo, On non-convex differential inclusions whose state is constrained in the closure of an open set, J.Differential Integral Equations, 12 (1999), 471-497.  Google Scholar

[5]

H. Frankowska and R. B. Vinter, Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 21-40.  Google Scholar

[6]

S. V. Grigor'eva, V. Y. Pakhotinskikh, A. A. Uspenskii and V. N. Ushakov, Construction of solutions in certain differential games with phase constraints, Sbornik Mathematics, 196 (2005), 513-539. Google Scholar

[7]

M. I. Gusev, On external estimates for reachable sets of nonlinear control systems, Proceedings of the Steklov Institute of Mathematics, 275, Suppl.1 (2011), 57-67. Google Scholar

[8]

M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Automation and Remote Control, 73 (2012), 450-461.  Google Scholar

[9]

M. I. Gusev, Internal approximations of reachable sets of control systems with state constraints, Proceedings of the Steklov Institute of Mathematics 287 (2014), 77-92. Google Scholar

[10]

A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems", Studies in Mathematics and its Applications, Amsterdam : North-Holland, 1979.  Google Scholar

[11]

E. K. Kostousova, On polyhedral estimates for reachable sets of multistep systems with bilinear uncertainty, Automation and Remote Control, 72 (2011), 1841-1851.  Google Scholar

[12]

A. B. Kurzhanski and T. F. Filippova, Description of the pencil of viable trajectories of a control system(Russian), Differentsial'nye Uravneniya, 23 (1987), 1303-1315.  Google Scholar

[13]

A. B. Kurzhanski, I. M. Mitchell and P. Varaiya, Optimization techniques for state-constrained control and obstacle problems, J. Optim. Theory Appl., 128 (2006), 499-521.  Google Scholar

[14]

A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control", SCFA. Boston: Birkhäuser, 1997.  Google Scholar

[15]

E. B. Lee and L. Markus, "Foundations of Optimal Control Theory", New York: Wiley, 1967.  Google Scholar

[16]

F. Lempio and V. M. Veliov, Discrete approximations of differential inclusions, GAMM Mitt. Ges. Angew. Math. Mech., 21 (1998), 103-135  Google Scholar

[17]

A.V. Lotov, A numerical method for constructing sets of attainability for linear controlled systems with phase constraints (Russian), Z. Vycisl. Mat. i Mat. Fiz, 15 (1975), 67-78.  Google Scholar

[18]

E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, System and Control Letters, 13 (1989), 117-123.  Google Scholar

[19]

R. J. Stern, Characterization of the State Constrained Minimal Time Function, SIAM J. Control and Optim. 43 (2004), 697-707.  Google Scholar

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