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On reachability analysis for nonlinear control systems with state constraints
1.  N.N.Krasovskii Institute of Mathematics and Mechanics, S.Kovalevskaya str., 16, 620099, Ekaterinburg, Russian Federation 
References:
[1] 
R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler method for stateconstrained differential inclusions, SIAM J. Optim., 18 (2007), 10041026. 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), 46644679. Google Scholar 
[3] 
N. Bonneuil, Computing reachable sets as captureviability kernels in reverse time, Applied Mathematics, 3 (2012), 15931597. Google Scholar 
[4] 
F. Forcellini and F. Rampazzo, On nonconvex differential inclusions whose state is constrained in the closure of an open set, J.Differential Integral Equations, 12 (1999), 471497. Google Scholar 
[5] 
H. Frankowska and R. B. Vinter, Existence of neighboring feasible trajectories: applications to dynamic programming for stateconstrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 2140. 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), 513539. 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), 5767. Google Scholar 
[8] 
M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Automation and Remote Control, 73 (2012), 450461. 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), 7792. Google Scholar 
[10] 
A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems", Studies in Mathematics and its Applications, Amsterdam : NorthHolland, 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), 18411851. 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), 13031315. Google Scholar 
[13] 
A. B. Kurzhanski, I. M. Mitchell and P. Varaiya, Optimization techniques for stateconstrained control and obstacle problems, J. Optim. Theory Appl., 128 (2006), 499521. 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), 103135 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), 6778. Google Scholar 
[18] 
E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, System and Control Letters, 13 (1989), 117123. Google Scholar 
[19] 
R. J. Stern, Characterization of the State Constrained Minimal Time Function, SIAM J. Control and Optim. 43 (2004), 697707. Google Scholar 
show all references
References:
[1] 
R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler method for stateconstrained differential inclusions, SIAM J. Optim., 18 (2007), 10041026. 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), 46644679. Google Scholar 
[3] 
N. Bonneuil, Computing reachable sets as captureviability kernels in reverse time, Applied Mathematics, 3 (2012), 15931597. Google Scholar 
[4] 
F. Forcellini and F. Rampazzo, On nonconvex differential inclusions whose state is constrained in the closure of an open set, J.Differential Integral Equations, 12 (1999), 471497. Google Scholar 
[5] 
H. Frankowska and R. B. Vinter, Existence of neighboring feasible trajectories: applications to dynamic programming for stateconstrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 2140. 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), 513539. 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), 5767. Google Scholar 
[8] 
M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Automation and Remote Control, 73 (2012), 450461. 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), 7792. Google Scholar 
[10] 
A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems", Studies in Mathematics and its Applications, Amsterdam : NorthHolland, 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), 18411851. 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), 13031315. Google Scholar 
[13] 
A. B. Kurzhanski, I. M. Mitchell and P. Varaiya, Optimization techniques for stateconstrained control and obstacle problems, J. Optim. Theory Appl., 128 (2006), 499521. 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), 103135 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), 6778. Google Scholar 
[18] 
E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, System and Control Letters, 13 (1989), 117123. Google Scholar 
[19] 
R. J. Stern, Characterization of the State Constrained Minimal Time Function, SIAM J. Control and Optim. 43 (2004), 697707. Google Scholar 
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