# American Institute of Mathematical Sciences

2015, 5(1): 1-9. doi: 10.3934/naco.2015.5.1

## Determining the viability for hybrid control systems on a region with piecewise smooth boundary

 1 School of Management, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai, 200093, China 2 Business School, University of Shanghai for Science and Technology, Shanghai, 200093

Received  December 2014 Revised  March 2015 Published  March 2015

This paper is devoted to determining the viability of hybrid control systems on a region which is expressed by inequalities of piecewise smooth functions. Firstly, the viability condition for the differential inclusion is discussed based on nonsmooth analysis. Secondly, the result is generalized to hybrid differential inclusion. Finally, the viability condition of differential inclusion on a region with the max-type function is given.
Citation: Yanli Han, Yan Gao. Determining the viability for hybrid control systems on a region with piecewise smooth boundary. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 1-9. doi: 10.3934/naco.2015.5.1
##### References:
 [1] P. J. Antsaklis and A. Nerode, Guest editorial hybrid control systems: an introductory discussion to the special issue,, IEEE Transactions on Automatic Control, 43 (1998), 457.   Google Scholar [2] J. P. Aubin, J. Lggeros, M. Quincampoix, S. Sastry and N. Seube, Impulse differential inclusions: A viability approach to hybrid systems,, IEEE Transactions on Automatic Control, 47 (2002), 2.  doi: 10.1109/9.981719.  Google Scholar [3] J. P. Aubin, Viability Theory,, Boston: Birkhauser, (1991).   Google Scholar [4] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis,, Springer, (1993).  doi: 10.1007/978-3-662-02959-6.  Google Scholar [5] F. Blanchini, Set invariance in control,, Automatica, 35 (1999), 1747.  doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar [6] R. W. Chaney, Piecewise $C^k$ functions in nonsmooth analysis,, Nonlinear Analysis, 15 (1990), 649.  doi: 10.1016/0362-546X(90)90005-2.  Google Scholar [7] F. H. Clark, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer-Verlag, (1998).   Google Scholar [8] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, Frankfurt am Main: Peterlang, (1995).   Google Scholar [9] Y. Gao, Determining the viability for an affine nonlinear control system (in Chinese),, Journal of Control Theory and Applications, 26 (2006), 654.   Google Scholar [10] Y. Gao, Determining the viability for a class of nonlinear control system on a region with nonsmooth boundary (in Chinese),, Control and Decision, 21 (2006), 923.   Google Scholar [11] Y. Gao, J. Lggeros, M. Quincampoix and N. Seube, On the control of uncertain impulsive system: approximate stabilization and controlled invariance,, International Journal of Control, 77 (2004), 1393.  doi: 10.1080/00207170412331317431.  Google Scholar [12] Y. Gao, J. Lggeros and M. Quincampoix, On the reachability problem of uncertain hybrid systems,, IEEE Transactions on Automatic Control, 52 (2007), 1572.  doi: 10.1109/TAC.2007.904449.  Google Scholar [13] Y. Gao, Nonsmooth Optimization (in Chinese),, Science Press, (2008).   Google Scholar [14] Y. Gao, Viability criteria for differential inclusions,, Journal of Systems Science Complexity, 24 (2011), 825.  doi: 10.1007/s11424-011-9056-6.  Google Scholar [15] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system,, Journal of Convex Analysis, 19 (2012), 1009.   Google Scholar [16] B. E. A. Milani and C. E. T. Dorea, On invariant polyhedra of continuous-time linear systems subject to additive disturbances,, Automatica, 32 (1996), 785.  doi: 10.1016/0005-1098(96)00002-7.  Google Scholar [17] B. Nikolai and T. Varvara, Numerical construction of viable sets for autonomous conflict control systems,, Mathematics, 2 (2014), 68.  doi: 10.3390/math2020068.  Google Scholar [18] P. S. Pierre, Hybrid Kernels and Capture Basins for Impulse Constrained Systems,, Proceedings of Hybrid Systems, (2003).   Google Scholar [19] M. Quincampoix and N. Seube, Stabilization of uncertain control systems through piecewise constant feedback,, Journal of Mathematical Analysis and Applications, 218 (1998), 240.  doi: 10.1006/jmaa.1997.5775.  Google Scholar

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##### References:
 [1] P. J. Antsaklis and A. Nerode, Guest editorial hybrid control systems: an introductory discussion to the special issue,, IEEE Transactions on Automatic Control, 43 (1998), 457.   Google Scholar [2] J. P. Aubin, J. Lggeros, M. Quincampoix, S. Sastry and N. Seube, Impulse differential inclusions: A viability approach to hybrid systems,, IEEE Transactions on Automatic Control, 47 (2002), 2.  doi: 10.1109/9.981719.  Google Scholar [3] J. P. Aubin, Viability Theory,, Boston: Birkhauser, (1991).   Google Scholar [4] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis,, Springer, (1993).  doi: 10.1007/978-3-662-02959-6.  Google Scholar [5] F. Blanchini, Set invariance in control,, Automatica, 35 (1999), 1747.  doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar [6] R. W. Chaney, Piecewise $C^k$ functions in nonsmooth analysis,, Nonlinear Analysis, 15 (1990), 649.  doi: 10.1016/0362-546X(90)90005-2.  Google Scholar [7] F. H. Clark, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer-Verlag, (1998).   Google Scholar [8] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, Frankfurt am Main: Peterlang, (1995).   Google Scholar [9] Y. Gao, Determining the viability for an affine nonlinear control system (in Chinese),, Journal of Control Theory and Applications, 26 (2006), 654.   Google Scholar [10] Y. Gao, Determining the viability for a class of nonlinear control system on a region with nonsmooth boundary (in Chinese),, Control and Decision, 21 (2006), 923.   Google Scholar [11] Y. Gao, J. Lggeros, M. Quincampoix and N. Seube, On the control of uncertain impulsive system: approximate stabilization and controlled invariance,, International Journal of Control, 77 (2004), 1393.  doi: 10.1080/00207170412331317431.  Google Scholar [12] Y. Gao, J. Lggeros and M. Quincampoix, On the reachability problem of uncertain hybrid systems,, IEEE Transactions on Automatic Control, 52 (2007), 1572.  doi: 10.1109/TAC.2007.904449.  Google Scholar [13] Y. Gao, Nonsmooth Optimization (in Chinese),, Science Press, (2008).   Google Scholar [14] Y. Gao, Viability criteria for differential inclusions,, Journal of Systems Science Complexity, 24 (2011), 825.  doi: 10.1007/s11424-011-9056-6.  Google Scholar [15] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system,, Journal of Convex Analysis, 19 (2012), 1009.   Google Scholar [16] B. E. A. Milani and C. E. T. Dorea, On invariant polyhedra of continuous-time linear systems subject to additive disturbances,, Automatica, 32 (1996), 785.  doi: 10.1016/0005-1098(96)00002-7.  Google Scholar [17] B. Nikolai and T. Varvara, Numerical construction of viable sets for autonomous conflict control systems,, Mathematics, 2 (2014), 68.  doi: 10.3390/math2020068.  Google Scholar [18] P. S. Pierre, Hybrid Kernels and Capture Basins for Impulse Constrained Systems,, Proceedings of Hybrid Systems, (2003).   Google Scholar [19] M. Quincampoix and N. Seube, Stabilization of uncertain control systems through piecewise constant feedback,, Journal of Mathematical Analysis and Applications, 218 (1998), 240.  doi: 10.1006/jmaa.1997.5775.  Google Scholar
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