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. ArutyunovD. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.  doi: 10.1007/s10958-010-9834-z.  Google Scholar

[2]

J.-P. Aubin and A. Cellina Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[3]

M. BranickyV. 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.  Google Scholar

[4]

B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer-Verlag, London, 1996.  Google Scholar

[5]

A. Bressan, On differential systems with impulsive controls, Rend. Sem. Math. Univ. Padova, 78 (1987), 227-235.   Google Scholar

[6]

A. Bressan, Hyperimpulsive motions and controllizable coordinates for Lagrangean systems, Atti. Acc. Lincei End. Fis, 19 (1989), 195-246.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.  Google Scholar

[12]

V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.   Google Scholar

[13]

V. Dykhta and O. Samsonyuk, Optimal'noe Impul'snoe Upravlenie s Prilozheniyami, (Russian) [Optimal impulsive control with applications], Fizmathlit, Moscow, 2000.  Google Scholar

[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.  Google Scholar

[15]

S. L. FragaR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21.   Google Scholar

[20]

V. Gurman, Singular Optimal Control Problems, Nauka, Moscow, 1977 (in Russian).  Google Scholar

[21]

W. Haddad, V. Chellaboina and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. Kurzhanski and P. Tochilin, Impulse controls in models of hybrid systems, Differential Equations, 45 (2009), 731-742.  doi: 10.1134/S0012266109050127.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128.  doi: 10.1016/0022-247X(62)90033-1.  Google Scholar

[33]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.  Google Scholar

[34]

J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser.A, 3 (1965), 424-438.  doi: 10.1137/0303028.  Google Scholar

[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.  Google Scholar

[36]

K. Yunt, Modelling of mechanical blocking, Recent Researches in Circuits, Systems, Mechanics and Transportation Systems, (2011), 123-128.   Google Scholar

show all references

References:
[1]

A. ArutyunovD. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.  doi: 10.1007/s10958-010-9834-z.  Google Scholar

[2]

J.-P. Aubin and A. Cellina Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[3]

M. BranickyV. 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.  Google Scholar

[4]

B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer-Verlag, London, 1996.  Google Scholar

[5]

A. Bressan, On differential systems with impulsive controls, Rend. Sem. Math. Univ. Padova, 78 (1987), 227-235.   Google Scholar

[6]

A. Bressan, Hyperimpulsive motions and controllizable coordinates for Lagrangean systems, Atti. Acc. Lincei End. Fis, 19 (1989), 195-246.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.  Google Scholar

[12]

V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.   Google Scholar

[13]

V. Dykhta and O. Samsonyuk, Optimal'noe Impul'snoe Upravlenie s Prilozheniyami, (Russian) [Optimal impulsive control with applications], Fizmathlit, Moscow, 2000.  Google Scholar

[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.  Google Scholar

[15]

S. L. FragaR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21.   Google Scholar

[20]

V. Gurman, Singular Optimal Control Problems, Nauka, Moscow, 1977 (in Russian).  Google Scholar

[21]

W. Haddad, V. Chellaboina and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. Kurzhanski and P. Tochilin, Impulse controls in models of hybrid systems, Differential Equations, 45 (2009), 731-742.  doi: 10.1134/S0012266109050127.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128.  doi: 10.1016/0022-247X(62)90033-1.  Google Scholar

[33]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.  Google Scholar

[34]

J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser.A, 3 (1965), 424-438.  doi: 10.1137/0303028.  Google Scholar

[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.  Google Scholar

[36]

K. Yunt, Modelling of mechanical blocking, Recent Researches in Circuits, Systems, Mechanics and Transportation Systems, (2011), 123-128.   Google Scholar

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