2013, 3(3): 519-548. doi: 10.3934/naco.2013.3.519

Approximation of reachable sets using optimal control algorithms

1. 

Applied Mathematics, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

2. 

Institute of Mathematics and Applied Computing (LRT), University of the Federal Armed Forces at Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

Received  October 2011 Revised  February 2013 Published  July 2013

We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence analysis is provided and shows the convergence of three different types of discrete set approximations to the reachable set. The distance functions can be computed numerically by suitable optimal control problems in combination with direct discretization techniques which allows adaptive calculations of reachable sets. Several numerical examples with nonconvex reachable sets are presented.
Citation: Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519
References:
[1]

H. Attouch and R. J.-B. Wets, Isometries for the Legendre-Fenchel transform,, Trans. Amer. Math. Soc., 296 (1986), 33.  doi: 10.1090/S0002-9947-1986-0837797-X.  Google Scholar

[2]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory. New Directions,", Second edition, (2011).  doi: 10.1007/978-3-642-16684-6.  Google Scholar

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R. Baier, "Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen,", Bayreuth. Math. Schr., 50 (1995).   Google Scholar

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R. Baier, Selection strategies for set-valued Runge-Kutta methods,, in, (2005), 149.   Google Scholar

[6]

R. Baier, Ch. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems,, Optim. Methods Softw., 22 (2007), 433.  doi: 10.1080/10556780600604999.  Google Scholar

[7]

R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions,, SIAM J. Optim., 18 (2007), 1004.  doi: 10.1137/060661867.  Google Scholar

[8]

R. Baier and M. Gerdts, A computational method for non-convex reachable sets using optimal control,, in, (2009), 23.   Google Scholar

[9]

W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91.  doi: 10.1007/s00607-007-0240-4.  Google Scholar

[10]

W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 14 (2010), 409.  doi: 10.3934/dcdsb.2010.14.409.  Google Scholar

[11]

H. G. Bock, "Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen,", Bonner Mathematische Schriften, 183 (1987).   Google Scholar

[12]

O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption,, SIAM J. Control Optim., 48 (2010), 4292.  doi: 10.1137/090762075.  Google Scholar

[13]

O. Bokanowski, A. Désilles, and H. Zidani, HJB approach for motion planning and reachabilty analysis,, in, (2011), 28.   Google Scholar

[14]

N. Bonneuil, Computing the viability kernel in large state dimension,, J. Math. Anal. Appl., 323 (2006), 1444.  doi: 10.1016/j.jmaa.2005.11.076.  Google Scholar

[15]

N. Bonneuil, Maximum under continuous-discrete-time dynamic with target and viability constraints,, Optimization, 61 (2012), 901.  doi: 10.1080/02331934.2011.605127.  Google Scholar

[16]

Y. Cao, S. Li, L. R. Petzold and R. Serban, Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,, SIAM J. Sci. Comput., 24 (2003), 1076.  doi: 10.1137/S1064827501380630.  Google Scholar

[17]

M. Caracotsios and W. E. Stewart, Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations,, Computers chem. Engng., 19 (1985), 1019.   Google Scholar

[18]

I. A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions,, Bayreuth. Math. Schr., 67 (2003), 3.   Google Scholar

[19]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).   Google Scholar

[20]

D. Cohen-Or, D. Levin and A. Solomovici, Three-dimensional distance field metamorphosis,, ACM Trans. Graph., 17 (1998), 116.   Google Scholar

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E. Crück, A. Désilles and H. Zidani, Collision analysis for an UAV,, in, (2012).   Google Scholar

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M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization,'', Second edition, (2011).   Google Scholar

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M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, 2 (2002), 221.   Google Scholar

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M. Dellnitz, O. Junge, M. Post and B. Thiere, On target for Venus-set oriented computation of energy efficient low thrust trajectories,, Celestial Mech. Dynam. Astronom., 95 (2006), 357.   Google Scholar

[25]

A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions,, Computing, 41 (1989), 349.   Google Scholar

[26]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order Runge-Kutta approximations in control constrained optimal control,, SIAM J. Numer. Anal., 38 (2000), 202.  doi: 10.1137/S0036142999351765.  Google Scholar

[27]

W. F. Feehery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems,, Appl. Numer. Math., 25 (1997), 41.  doi: 10.1016/S0168-9274(97)00050-0.  Google Scholar

[28]

T. F. Filippova and E. V. Berezina, On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations,, in, (2008), 326.   Google Scholar

[29]

H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wa.zewski's theorems on closed domains,, J. Differ. Equ., 161 (2000), 449.  doi: 10.1006/jdeq.2000.3711.  Google Scholar

[30]

J. E. Gayek, Approximating reachable sets for a class of linear control systems,, Internat. J. Control, 43 (1986), 441.  doi: 10.1080/00207178608933477.  Google Scholar

[31]

M. Gerdts, "User manual for OCPID-DAE1,", User manual, (2010).   Google Scholar

[32]

M. Gerdts, "Optimal Control of ODEs and DAEs,", DeGruyter, (2011).   Google Scholar

[33]

A. Girard and C. Le Guernic, Zonotope/hyperplane intersection for hybrid systems reachability analysis,, in, (2008), 22.   Google Scholar

[34]

A. Griewank, "Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation,", volume 19 of, (2000).   Google Scholar

[35]

L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'', volume 1783 of, (1783).   Google Scholar

[36]

G. Häckl, "Reachable Sets, Control Sets and Their Computation. With a Preface by F. Colonius,", volume 7 of, (1995).   Google Scholar

[37]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247.  doi: 10.1007/s002110000178.  Google Scholar

[38]

O. Hájek, "Control Theory in the Plane,'', Second edition, (2008).   Google Scholar

[39]

H. Hermes and J. P. Lasalle, Functional Analysis and Time Optimal Control,, in, (1969).   Google Scholar

[40]

P. Kenderov, Dense strong continuity of pointwise continuous mappings,, Pacific J. Math., 89 (1980), 111.  doi: 10.2140/pjm.1980.89.111.  Google Scholar

[41]

N. Kirov and M. Krastanov, Volterra series and numerical approximations of ODEs,, in, 3401 (2005), 337.   Google Scholar

[42]

E. K. Kostousova, State estimation for dynamic systems via parallelotopes: optimization and parallel computations,, Optim. Methods Softw., 9 (1998), 269.  doi: 10.1080/10556789808805696.  Google Scholar

[43]

E. K. Kostousova, State estimation for control systems with a multiplicative uncertainty through polyhedral techniques,, in, (2012), 12.   Google Scholar

[44]

M. I. Krastanov and V. M. Veliov, High-order approximations to nonholonomic affine control systems,, in, (2010), 4.   Google Scholar

[45]

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.  doi: 10.1007/s10957-006-9029-4.  Google Scholar

[46]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability analysis: internal approximation,, Systems Control Lett., 41 (2000), 201.  doi: 10.1016/S0167-6911(00)00059-1.  Google Scholar

[47]

A. B. Kurzhanski and P. Varaiya, Dynamic optimization for reachability problems,, J. Optim. Theory Appl., 108 (2001), 227.  doi: 10.1023/A:1026497115405.  Google Scholar

[48]

A. B. Kurzhanski and P. Varaiya, On ellipsoidal techniques for reachability analysis. Part I: external approximations,, Optim. Methods Softw., 17 (2002), 177.  doi: 10.1080/1055678021000012426.  Google Scholar

[49]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability under state constraints,, SIAM J. Control Optim., 45 (2006), 1369.  doi: 10.1137/S0363012903437605.  Google Scholar

[50]

D. Levin, Multidimensional reconstruction by set-valued approximations,, IMA J. Numer. Anal., 6 (1986), 173.  doi: 10.1093/imanum/6.2.173.  Google Scholar

[51]

T. Lorenz, Epi-Lipschitzian reachable sets of differential inclusions,, Syst. Control Lett., 57 (2008), 703.  doi: 10.1016/j.sysconle.2008.01.007.  Google Scholar

[52]

K. Malanowski, Ch. Büskens, and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1997), 253.   Google Scholar

[53]

T. Maly and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems,, Appl. Numer. Math., 20 (1996), 57.  doi: 10.1016/0168-9274(95)00117-4.  Google Scholar

[54]

I. M. Mitchell, Comparing forward and backward reachability as tools for safety analysis,, in, (2007), 428.   Google Scholar

[55]

I. M. Mitchell and C. J. Tomlin, Overapproximating reachable sets by Hamilton-Jacobi projections,, J. Sci. Comput., 19 (2003), 323.  doi: 10.1023/A:1025364227563.  Google Scholar

[56]

J. Nocedal and S. J. Wright, "Numerical Optimization,'', Springer Series in Operations Research, (1999).  doi: 10.1007/b98874.  Google Scholar

[57]

A. Pietrus and V. M. Veliov, On the discretization of switched linear systems,, System Control Lett., 58 (2009), 395.  doi: 10.1016/j.sysconle.2009.01.005.  Google Scholar

[58]

A. Puri, V. Borkar, and P. Varaiya, $\epsilon$-Approximations of differential inclusions,, in, (1996), 20.   Google Scholar

[59]

M. Quincampoix and V. M. Veliov, Optimal control of uncertain systems with incomplete information for the disturbances,, SIAM J. Control Optim., 43 (): 1373.  doi: 10.1137/S0363012903420863.  Google Scholar

[60]

J. Rieger, Shadowing and the viability kernel algorithm,, Appl. Math. Optim., 60 (2009), 429.  doi: 10.1007/s00245-009-9083-z.  Google Scholar

[61]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,'', volume 317 of, (1998).   Google Scholar

[62]

P. Saint-Pierre, Approximation of the viability kernel,, Appl. Math. Optim., 29 (1994), 187.  doi: 10.1007/BF01204182.  Google Scholar

[63]

M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions,, SIAM J. Numer. Anal., 47 (): 308.  doi: 10.1137/070686093.  Google Scholar

[64]

P. Varaiya, Reach set computation using optimal control,, in, (2000), 323.   Google Scholar

[65]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions,, Systems Control Lett., 13 (1989), 263.  doi: 10.1016/0167-6911(89)90073-X.  Google Scholar

[66]

V. M. Veliov, Second order discrete approximation to linear differential inclusions,, SIAM J. Numer. Anal., 29 (1992), 439.  doi: 10.1137/0729026.  Google Scholar

[67]

P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion,, SIAM J. Control Optim., 28 (1990), 1148.  doi: 10.1137/0328062.  Google Scholar

show all references

References:
[1]

H. Attouch and R. J.-B. Wets, Isometries for the Legendre-Fenchel transform,, Trans. Amer. Math. Soc., 296 (1986), 33.  doi: 10.1090/S0002-9947-1986-0837797-X.  Google Scholar

[2]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory. New Directions,", Second edition, (2011).  doi: 10.1007/978-3-642-16684-6.  Google Scholar

[3]

J.-P. Aubin, T. Bernado and P. Saint-Pierre, A viability approach to global climate change issues,, in, (2005), 113.   Google Scholar

[4]

R. Baier, "Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen,", Bayreuth. Math. Schr., 50 (1995).   Google Scholar

[5]

R. Baier, Selection strategies for set-valued Runge-Kutta methods,, in, (2005), 149.   Google Scholar

[6]

R. Baier, Ch. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems,, Optim. Methods Softw., 22 (2007), 433.  doi: 10.1080/10556780600604999.  Google Scholar

[7]

R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions,, SIAM J. Optim., 18 (2007), 1004.  doi: 10.1137/060661867.  Google Scholar

[8]

R. Baier and M. Gerdts, A computational method for non-convex reachable sets using optimal control,, in, (2009), 23.   Google Scholar

[9]

W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91.  doi: 10.1007/s00607-007-0240-4.  Google Scholar

[10]

W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 14 (2010), 409.  doi: 10.3934/dcdsb.2010.14.409.  Google Scholar

[11]

H. G. Bock, "Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen,", Bonner Mathematische Schriften, 183 (1987).   Google Scholar

[12]

O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption,, SIAM J. Control Optim., 48 (2010), 4292.  doi: 10.1137/090762075.  Google Scholar

[13]

O. Bokanowski, A. Désilles, and H. Zidani, HJB approach for motion planning and reachabilty analysis,, in, (2011), 28.   Google Scholar

[14]

N. Bonneuil, Computing the viability kernel in large state dimension,, J. Math. Anal. Appl., 323 (2006), 1444.  doi: 10.1016/j.jmaa.2005.11.076.  Google Scholar

[15]

N. Bonneuil, Maximum under continuous-discrete-time dynamic with target and viability constraints,, Optimization, 61 (2012), 901.  doi: 10.1080/02331934.2011.605127.  Google Scholar

[16]

Y. Cao, S. Li, L. R. Petzold and R. Serban, Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,, SIAM J. Sci. Comput., 24 (2003), 1076.  doi: 10.1137/S1064827501380630.  Google Scholar

[17]

M. Caracotsios and W. E. Stewart, Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations,, Computers chem. Engng., 19 (1985), 1019.   Google Scholar

[18]

I. A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions,, Bayreuth. Math. Schr., 67 (2003), 3.   Google Scholar

[19]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).   Google Scholar

[20]

D. Cohen-Or, D. Levin and A. Solomovici, Three-dimensional distance field metamorphosis,, ACM Trans. Graph., 17 (1998), 116.   Google Scholar

[21]

E. Crück, A. Désilles and H. Zidani, Collision analysis for an UAV,, in, (2012).   Google Scholar

[22]

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization,'', Second edition, (2011).   Google Scholar

[23]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, 2 (2002), 221.   Google Scholar

[24]

M. Dellnitz, O. Junge, M. Post and B. Thiere, On target for Venus-set oriented computation of energy efficient low thrust trajectories,, Celestial Mech. Dynam. Astronom., 95 (2006), 357.   Google Scholar

[25]

A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions,, Computing, 41 (1989), 349.   Google Scholar

[26]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order Runge-Kutta approximations in control constrained optimal control,, SIAM J. Numer. Anal., 38 (2000), 202.  doi: 10.1137/S0036142999351765.  Google Scholar

[27]

W. F. Feehery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems,, Appl. Numer. Math., 25 (1997), 41.  doi: 10.1016/S0168-9274(97)00050-0.  Google Scholar

[28]

T. F. Filippova and E. V. Berezina, On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations,, in, (2008), 326.   Google Scholar

[29]

H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wa.zewski's theorems on closed domains,, J. Differ. Equ., 161 (2000), 449.  doi: 10.1006/jdeq.2000.3711.  Google Scholar

[30]

J. E. Gayek, Approximating reachable sets for a class of linear control systems,, Internat. J. Control, 43 (1986), 441.  doi: 10.1080/00207178608933477.  Google Scholar

[31]

M. Gerdts, "User manual for OCPID-DAE1,", User manual, (2010).   Google Scholar

[32]

M. Gerdts, "Optimal Control of ODEs and DAEs,", DeGruyter, (2011).   Google Scholar

[33]

A. Girard and C. Le Guernic, Zonotope/hyperplane intersection for hybrid systems reachability analysis,, in, (2008), 22.   Google Scholar

[34]

A. Griewank, "Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation,", volume 19 of, (2000).   Google Scholar

[35]

L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'', volume 1783 of, (1783).   Google Scholar

[36]

G. Häckl, "Reachable Sets, Control Sets and Their Computation. With a Preface by F. Colonius,", volume 7 of, (1995).   Google Scholar

[37]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247.  doi: 10.1007/s002110000178.  Google Scholar

[38]

O. Hájek, "Control Theory in the Plane,'', Second edition, (2008).   Google Scholar

[39]

H. Hermes and J. P. Lasalle, Functional Analysis and Time Optimal Control,, in, (1969).   Google Scholar

[40]

P. Kenderov, Dense strong continuity of pointwise continuous mappings,, Pacific J. Math., 89 (1980), 111.  doi: 10.2140/pjm.1980.89.111.  Google Scholar

[41]

N. Kirov and M. Krastanov, Volterra series and numerical approximations of ODEs,, in, 3401 (2005), 337.   Google Scholar

[42]

E. K. Kostousova, State estimation for dynamic systems via parallelotopes: optimization and parallel computations,, Optim. Methods Softw., 9 (1998), 269.  doi: 10.1080/10556789808805696.  Google Scholar

[43]

E. K. Kostousova, State estimation for control systems with a multiplicative uncertainty through polyhedral techniques,, in, (2012), 12.   Google Scholar

[44]

M. I. Krastanov and V. M. Veliov, High-order approximations to nonholonomic affine control systems,, in, (2010), 4.   Google Scholar

[45]

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.  doi: 10.1007/s10957-006-9029-4.  Google Scholar

[46]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability analysis: internal approximation,, Systems Control Lett., 41 (2000), 201.  doi: 10.1016/S0167-6911(00)00059-1.  Google Scholar

[47]

A. B. Kurzhanski and P. Varaiya, Dynamic optimization for reachability problems,, J. Optim. Theory Appl., 108 (2001), 227.  doi: 10.1023/A:1026497115405.  Google Scholar

[48]

A. B. Kurzhanski and P. Varaiya, On ellipsoidal techniques for reachability analysis. Part I: external approximations,, Optim. Methods Softw., 17 (2002), 177.  doi: 10.1080/1055678021000012426.  Google Scholar

[49]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability under state constraints,, SIAM J. Control Optim., 45 (2006), 1369.  doi: 10.1137/S0363012903437605.  Google Scholar

[50]

D. Levin, Multidimensional reconstruction by set-valued approximations,, IMA J. Numer. Anal., 6 (1986), 173.  doi: 10.1093/imanum/6.2.173.  Google Scholar

[51]

T. Lorenz, Epi-Lipschitzian reachable sets of differential inclusions,, Syst. Control Lett., 57 (2008), 703.  doi: 10.1016/j.sysconle.2008.01.007.  Google Scholar

[52]

K. Malanowski, Ch. Büskens, and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1997), 253.   Google Scholar

[53]

T. Maly and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems,, Appl. Numer. Math., 20 (1996), 57.  doi: 10.1016/0168-9274(95)00117-4.  Google Scholar

[54]

I. M. Mitchell, Comparing forward and backward reachability as tools for safety analysis,, in, (2007), 428.   Google Scholar

[55]

I. M. Mitchell and C. J. Tomlin, Overapproximating reachable sets by Hamilton-Jacobi projections,, J. Sci. Comput., 19 (2003), 323.  doi: 10.1023/A:1025364227563.  Google Scholar

[56]

J. Nocedal and S. J. Wright, "Numerical Optimization,'', Springer Series in Operations Research, (1999).  doi: 10.1007/b98874.  Google Scholar

[57]

A. Pietrus and V. M. Veliov, On the discretization of switched linear systems,, System Control Lett., 58 (2009), 395.  doi: 10.1016/j.sysconle.2009.01.005.  Google Scholar

[58]

A. Puri, V. Borkar, and P. Varaiya, $\epsilon$-Approximations of differential inclusions,, in, (1996), 20.   Google Scholar

[59]

M. Quincampoix and V. M. Veliov, Optimal control of uncertain systems with incomplete information for the disturbances,, SIAM J. Control Optim., 43 (): 1373.  doi: 10.1137/S0363012903420863.  Google Scholar

[60]

J. Rieger, Shadowing and the viability kernel algorithm,, Appl. Math. Optim., 60 (2009), 429.  doi: 10.1007/s00245-009-9083-z.  Google Scholar

[61]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,'', volume 317 of, (1998).   Google Scholar

[62]

P. Saint-Pierre, Approximation of the viability kernel,, Appl. Math. Optim., 29 (1994), 187.  doi: 10.1007/BF01204182.  Google Scholar

[63]

M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions,, SIAM J. Numer. Anal., 47 (): 308.  doi: 10.1137/070686093.  Google Scholar

[64]

P. Varaiya, Reach set computation using optimal control,, in, (2000), 323.   Google Scholar

[65]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions,, Systems Control Lett., 13 (1989), 263.  doi: 10.1016/0167-6911(89)90073-X.  Google Scholar

[66]

V. M. Veliov, Second order discrete approximation to linear differential inclusions,, SIAM J. Numer. Anal., 29 (1992), 439.  doi: 10.1137/0729026.  Google Scholar

[67]

P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion,, SIAM J. Control Optim., 28 (1990), 1148.  doi: 10.1137/0328062.  Google Scholar

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