# American Institute of Mathematical Sciences

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

## 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

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.
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. [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. [3] N. Bonneuil, Computing reachable sets as capture-viability kernels in reverse time, Applied Mathematics, 3 (2012), 1593-1597. [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. [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. [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. [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. [8] M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Automation and Remote Control, 73 (2012), 450-461. [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. [10] A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems", Studies in Mathematics and its Applications, Amsterdam : North-Holland, 1979. [11] E. K. Kostousova, On polyhedral estimates for reachable sets of multistep systems with bilinear uncertainty, Automation and Remote Control, 72 (2011), 1841-1851. [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. [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. [14] A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control", SCFA. Boston: Birkhäuser, 1997. [15] E. B. Lee and L. Markus, "Foundations of Optimal Control Theory", New York: Wiley, 1967. [16] F. Lempio and V. M. Veliov, Discrete approximations of differential inclusions, GAMM Mitt. Ges. Angew. Math. Mech., 21 (1998), 103-135 [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. [18] E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, System and Control Letters, 13 (1989), 117-123. [19] R. J. Stern, Characterization of the State Constrained Minimal Time Function, SIAM J. Control and Optim. 43 (2004), 697-707.

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. [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. [3] N. Bonneuil, Computing reachable sets as capture-viability kernels in reverse time, Applied Mathematics, 3 (2012), 1593-1597. [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. [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. [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. [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. [8] M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Automation and Remote Control, 73 (2012), 450-461. [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. [10] A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems", Studies in Mathematics and its Applications, Amsterdam : North-Holland, 1979. [11] E. K. Kostousova, On polyhedral estimates for reachable sets of multistep systems with bilinear uncertainty, Automation and Remote Control, 72 (2011), 1841-1851. [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. [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. [14] A. B. Kurzhanski and I. Valyi, "Ellipsoidal Calculus for Estimation and Control", SCFA. Boston: Birkhäuser, 1997. [15] E. B. Lee and L. Markus, "Foundations of Optimal Control Theory", New York: Wiley, 1967. [16] F. Lempio and V. M. Veliov, Discrete approximations of differential inclusions, GAMM Mitt. Ges. Angew. Math. Mech., 21 (1998), 103-135 [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. [18] E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, System and Control Letters, 13 (1989), 117-123. [19] R. J. Stern, Characterization of the State Constrained Minimal Time Function, SIAM J. Control and Optim. 43 (2004), 697-707.
 [1] Jianling Li, Chunting Lu, Youfang Zeng. A smooth QP-free algorithm without a penalty function or a filter for mathematical programs with complementarity constraints. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 115-126. doi: 10.3934/naco.2015.5.115 [2] Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial and Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533 [3] Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 [4] Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041 [5] Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109 [6] Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361 [7] Canghua Jiang, Dongming Zhang, Chi Yuan, Kok Ley Teo. An active set solver for constrained $H_\infty$ optimal control problems with state and input constraints. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 135-157. doi: 10.3934/naco.2021056 [8] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial and Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [9] H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181 [10] Anna-Lena Horlemann-Trautmann, Violetta Weger. Information set decoding in the Lee metric with applications to cryptography. Advances in Mathematics of Communications, 2021, 15 (4) : 677-699. doi: 10.3934/amc.2020089 [11] Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022, 27 (11) : 6589-6604. doi: 10.3934/dcdsb.2022010 [12] Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076 [13] Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933 [14] X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial and Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287 [15] Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial and Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585 [16] Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial and Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 [17] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895 [18] Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017 [19] Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations and Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173 [20] Jean-François Babadjian, Clément Mifsud, Nicolas Seguin. Relaxation approximation of Friedrichs' systems under convex constraints. Networks and Heterogeneous Media, 2016, 11 (2) : 223-237. doi: 10.3934/nhm.2016.11.223

Impact Factor: