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Noncontrollability for the ColemannGurtin model in several dimensions
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), 1004. 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. Google Scholar 
[3] 
N. Bonneuil, Computing reachable sets as captureviability kernels in reverse time,, Applied Mathematics, 3 (2012), 1593. 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), 471. 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), 21. 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. Google Scholar 
[7] 
M. I. Gusev, On external estimates for reachable sets of nonlinear control systems,, Proceedings of the Steklov Institute of Mathematics, 275 (2011), 57. Google Scholar 
[8] 
M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems,, Automation and Remote Control, 73 (2012), 450. 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), 287 (2014), 77. Google Scholar 
[10] 
A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems",, Studies in Mathematics and its Applications, (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. 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. 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), 499. 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. 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. Google Scholar 
[18] 
E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization,, System and Control Letters, 13 (1989), 117. Google Scholar 
[19] 
R. J. Stern, Characterization of the State Constrained Minimal Time Function,, SIAM J. Control and Optim. 43 (2004), 43 (2004), 697. 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), 1004. 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. Google Scholar 
[3] 
N. Bonneuil, Computing reachable sets as captureviability kernels in reverse time,, Applied Mathematics, 3 (2012), 1593. 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), 471. 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), 21. 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. Google Scholar 
[7] 
M. I. Gusev, On external estimates for reachable sets of nonlinear control systems,, Proceedings of the Steklov Institute of Mathematics, 275 (2011), 57. Google Scholar 
[8] 
M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems,, Automation and Remote Control, 73 (2012), 450. 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), 287 (2014), 77. Google Scholar 
[10] 
A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems",, Studies in Mathematics and its Applications, (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. 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. 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), 499. 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. 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. Google Scholar 
[18] 
E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization,, System and Control Letters, 13 (1989), 117. Google Scholar 
[19] 
R. J. Stern, Characterization of the State Constrained Minimal Time Function,, SIAM J. Control and Optim. 43 (2004), 43 (2004), 697. Google Scholar 
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