# American Institute of Mathematical Sciences

December 2018, 11(6): 1061-1070. doi: 10.3934/dcdss.2018061

## On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations

 Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia

Received  March 2017 Revised  June 2017 Published  June 2018

Fund Project: The work was partially supported by the Russian Foundation for Basic Research, projects 16-08-00272, 16-31-60030, 16-31-00184, 17-08-00742

The goal of the paper is to design a constructive impulsive trajectory extension for a class of control-affine dynamical systems subject to a asymptotic mixed constraint of complementarity type. An inspiration for the addressed models comes from the framework of Lagrangian mechanical systems with impactively blockable degrees of freedom. The constraint formalizes the requirement that "control actions steer the system's state from one prescribed configuration $\mathcal{Z}_-$ to another one $\mathcal{Z}_+$". This issue is also closely connected with the problem of continuous trajectory approximation of hybrid systems with control switches.

Citation: Elena Goncharova, Maxim Staritsyn. On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1061-1070. doi: 10.3934/dcdss.2018061
##### References:
 [1] A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688. doi: 10.1007/s10958-010-9834-z. [2] J.-P. Aubin and A. Cellina Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [3] M. Branicky, V. Borkar and S. Mitter, A unified framework for hybrid control: Model and optimal control theory, IEEE Trans. Automat. Control, 43 (1998), 31-45. doi: 10.1109/9.654885. [4] B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer-Verlag, London, 1996. [5] A. Bressan, On differential systems with impulsive controls, Rend. Sem. Math. Univ. Padova, 78 (1987), 227-235. [6] A. Bressan, Hyperimpulsive motions and controllizable coordinates for Lagrangean systems, Atti. Acc. Lincei End. Fis, 19 (1989), 195-246. [7] A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1. [8] A. Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993), 30pp. [9] A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6. [10] A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics, SIAM J. Control Optim., 31 (1993), 1205-1220. doi: 10.1137/0331057. [11] A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094. [12] V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109. [13] V. Dykhta and O. Samsonyuk, Optimal'noe Impul'snoe Upravlenie s Prilozheniyami, (Russian) [Optimal impulsive control with applications], Fizmathlit, Moscow, 2000. [14] V. Dykhta and O. Samsonyuk, A maximum principle for smooth optimal impulsive control problems with multipoint state constraints, Comput. Math. Math. Phys., 49 (2009), 942-957. doi: 10.1134/S0965542509060050. [15] S. L. Fraga, R. Gomes and F. L. Pereira, An impulsive framework for the control of hybrid systems, Proc. 46 IEEE Conf. Decision Control, (2007), 5444-5449. doi: 10.1109/CDC.2007.4434895. [16] Ch. Glocker, Impacts with global dissipation index at reentrant corners, in Contact Mechanics (eds. J. A. C. Martins and M. D. P. Monteiro Marques), Springer, 103 (2002), 45–52. doi: 10.1007/978-94-017-1154-8_5. [17] E. Goncharova and M. Staritsyn, Optimization of measure-driven hybrid systems, J. Optim. Theory Appl., 153 (2012), 139-156. doi: 10.1007/s10957-011-9944-x. [18] E. Goncharova and M. Staritsyn, Optimal impulsive control problem with state and mixed constraints: The case of vector-valued measure, Autom. Rem. Control, 76 (2015), 377-387. doi: 10.1134/S0005117915030029. [19] V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21. [20] V. Gurman, Singular Optimal Control Problems, Nauka, Moscow, 1977 (in Russian). [21] W. Haddad, V. Chellaboina and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006. [22] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, Journal of Differential Equations, 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711. [23] D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem, J. Math. Sci., 139 (2006), 7087-7150. doi: 10.1007/s10958-006-0408-z. [24] A. Kurzhanski and P. Tochilin, Impulse controls in models of hybrid systems, Differential Equations, 45 (2009), 731-742. doi: 10.1134/S0012266109050127. [25] H. O. May, Generalized variational principles and unilateral constraints in analytical mechanics, in Unilateral Problems in Structural Analysis-2: Proc. 2nd Meeting on Unilateral Problems in Structural Analysis, (1987), 221–237. doi: 10.1007/978-3-7091-2967-8_12. [26] B. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214. [27] B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7. [28] B. Miller and J. Bentsman, Optimal control problems in hybrid systems with active singularities, Nonlinear Analysis, 65 (2006), 999-1017. doi: 10.1016/j.na.2005.08.033. [29] M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. [30] R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. doi: 10.1137/0303016. [31] R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229. doi: 10.1137/0326013. [32] J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1. [33] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. [34] J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser.A, 3 (1965), 424-438. doi: 10.1137/0303028. [35] S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5. [36] K. Yunt, Modelling of mechanical blocking, Recent Researches in Circuits, Systems, Mechanics and Transportation Systems, (2011), 123-128.

show all references

##### References:
 [1] A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688. doi: 10.1007/s10958-010-9834-z. [2] J.-P. Aubin and A. Cellina Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [3] M. Branicky, V. Borkar and S. Mitter, A unified framework for hybrid control: Model and optimal control theory, IEEE Trans. Automat. Control, 43 (1998), 31-45. doi: 10.1109/9.654885. [4] B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer-Verlag, London, 1996. [5] A. Bressan, On differential systems with impulsive controls, Rend. Sem. Math. Univ. Padova, 78 (1987), 227-235. [6] A. Bressan, Hyperimpulsive motions and controllizable coordinates for Lagrangean systems, Atti. Acc. Lincei End. Fis, 19 (1989), 195-246. [7] A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1. [8] A. Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993), 30pp. [9] A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6. [10] A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics, SIAM J. Control Optim., 31 (1993), 1205-1220. doi: 10.1137/0331057. [11] A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094. [12] V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109. [13] V. Dykhta and O. Samsonyuk, Optimal'noe Impul'snoe Upravlenie s Prilozheniyami, (Russian) [Optimal impulsive control with applications], Fizmathlit, Moscow, 2000. [14] V. Dykhta and O. Samsonyuk, A maximum principle for smooth optimal impulsive control problems with multipoint state constraints, Comput. Math. Math. Phys., 49 (2009), 942-957. doi: 10.1134/S0965542509060050. [15] S. L. Fraga, R. Gomes and F. L. Pereira, An impulsive framework for the control of hybrid systems, Proc. 46 IEEE Conf. Decision Control, (2007), 5444-5449. doi: 10.1109/CDC.2007.4434895. [16] Ch. Glocker, Impacts with global dissipation index at reentrant corners, in Contact Mechanics (eds. J. A. C. Martins and M. D. P. Monteiro Marques), Springer, 103 (2002), 45–52. doi: 10.1007/978-94-017-1154-8_5. [17] E. Goncharova and M. Staritsyn, Optimization of measure-driven hybrid systems, J. Optim. Theory Appl., 153 (2012), 139-156. doi: 10.1007/s10957-011-9944-x. [18] E. Goncharova and M. Staritsyn, Optimal impulsive control problem with state and mixed constraints: The case of vector-valued measure, Autom. Rem. Control, 76 (2015), 377-387. doi: 10.1134/S0005117915030029. [19] V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21. [20] V. Gurman, Singular Optimal Control Problems, Nauka, Moscow, 1977 (in Russian). [21] W. Haddad, V. Chellaboina and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006. [22] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, Journal of Differential Equations, 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711. [23] D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem, J. Math. Sci., 139 (2006), 7087-7150. doi: 10.1007/s10958-006-0408-z. [24] A. Kurzhanski and P. Tochilin, Impulse controls in models of hybrid systems, Differential Equations, 45 (2009), 731-742. doi: 10.1134/S0012266109050127. [25] H. O. May, Generalized variational principles and unilateral constraints in analytical mechanics, in Unilateral Problems in Structural Analysis-2: Proc. 2nd Meeting on Unilateral Problems in Structural Analysis, (1987), 221–237. doi: 10.1007/978-3-7091-2967-8_12. [26] B. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214. [27] B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7. [28] B. Miller and J. Bentsman, Optimal control problems in hybrid systems with active singularities, Nonlinear Analysis, 65 (2006), 999-1017. doi: 10.1016/j.na.2005.08.033. [29] M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. [30] R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. doi: 10.1137/0303016. [31] R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229. doi: 10.1137/0326013. [32] J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1. [33] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. [34] J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser.A, 3 (1965), 424-438. doi: 10.1137/0303028. [35] S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5. [36] K. Yunt, Modelling of mechanical blocking, Recent Researches in Circuits, Systems, Mechanics and Transportation Systems, (2011), 123-128.
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