# 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 and 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), 457-460. [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), 2-20. doi: 10.1109/9.981719. [3] J. P. Aubin, Viability Theory, Boston: Birkhauser, 1991. [4] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02959-6. [5] F. Blanchini, Set invariance in control, Automatica, 35 (1999), 1747-1767. doi: 10.1016/S0005-1098(99)00113-2. [6] R. W. Chaney, Piecewise $C^k$ functions in nonsmooth analysis, Nonlinear Analysis, 15 (1990), 649-660. doi: 10.1016/0362-546X(90)90005-2. [7] F. H. Clark, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. [8] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Frankfurt am Main: Peterlang, 1995. [9] Y. Gao, Determining the viability for an affine nonlinear control system (in Chinese), Journal of Control Theory and Applications, 26 (2006), 654-656. [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), 923-925. [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), 1393-1407. doi: 10.1080/00207170412331317431. [12] Y. Gao, J. Lggeros and M. Quincampoix, On the reachability problem of uncertain hybrid systems, IEEE Transactions on Automatic Control, 52 (2007), 1572-1586. doi: 10.1109/TAC.2007.904449. [13] Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008. [14] Y. Gao, Viability criteria for differential inclusions, Journal of Systems Science Complexity, 24 (2011), 825-834. doi: 10.1007/s11424-011-9056-6. [15] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, Journal of Convex Analysis, 19 (2012), 1009-1015. [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), 785-789. doi: 10.1016/0005-1098(96)00002-7. [17] B. Nikolai and T. Varvara, Numerical construction of viable sets for autonomous conflict control systems, Mathematics, 2 (2014), 68-82. doi: 10.3390/math2020068. [18] P. S. Pierre, Hybrid Kernels and Capture Basins for Impulse Constrained Systems, Proceedings of Hybrid Systems, 2003. [19] M. Quincampoix and N. Seube, Stabilization of uncertain control systems through piecewise constant feedback, Journal of Mathematical Analysis and Applications, 218 (1998), 240-255. doi: 10.1006/jmaa.1997.5775.

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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), 457-460. [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), 2-20. doi: 10.1109/9.981719. [3] J. P. Aubin, Viability Theory, Boston: Birkhauser, 1991. [4] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02959-6. [5] F. Blanchini, Set invariance in control, Automatica, 35 (1999), 1747-1767. doi: 10.1016/S0005-1098(99)00113-2. [6] R. W. Chaney, Piecewise $C^k$ functions in nonsmooth analysis, Nonlinear Analysis, 15 (1990), 649-660. doi: 10.1016/0362-546X(90)90005-2. [7] F. H. Clark, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. [8] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Frankfurt am Main: Peterlang, 1995. [9] Y. Gao, Determining the viability for an affine nonlinear control system (in Chinese), Journal of Control Theory and Applications, 26 (2006), 654-656. [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), 923-925. [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), 1393-1407. doi: 10.1080/00207170412331317431. [12] Y. Gao, J. Lggeros and M. Quincampoix, On the reachability problem of uncertain hybrid systems, IEEE Transactions on Automatic Control, 52 (2007), 1572-1586. doi: 10.1109/TAC.2007.904449. [13] Y. Gao, Nonsmooth Optimization (in Chinese), Science Press, Beijing, 2008. [14] Y. Gao, Viability criteria for differential inclusions, Journal of Systems Science Complexity, 24 (2011), 825-834. doi: 10.1007/s11424-011-9056-6. [15] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, Journal of Convex Analysis, 19 (2012), 1009-1015. [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), 785-789. doi: 10.1016/0005-1098(96)00002-7. [17] B. Nikolai and T. Varvara, Numerical construction of viable sets for autonomous conflict control systems, Mathematics, 2 (2014), 68-82. doi: 10.3390/math2020068. [18] P. S. Pierre, Hybrid Kernels and Capture Basins for Impulse Constrained Systems, Proceedings of Hybrid Systems, 2003. [19] M. Quincampoix and N. Seube, Stabilization of uncertain control systems through piecewise constant feedback, Journal of Mathematical Analysis and Applications, 218 (1998), 240-255. doi: 10.1006/jmaa.1997.5775.
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