November 2018, 38(11): 5781-5809. doi: 10.3934/dcds.2018252

Minimum time problem with impulsive and ordinary controls

Università di Padova, Via Trieste, 63, Padova 35121, Italy

Received  January 2018 Revised  April 2018 Published  August 2018

Fund Project: This research is partially supported by the INdAM-GNAMPA Project 2017 "Optimal impulsive control: higher order necessary conditions and gap phenomena"; and by the Padova University grant PRAT 2015 "Control of dynamics with reactive constraints"

Given a nonlinear control system depending on two controls $u$ and $v$, with dynamics affine in the (unbounded) derivative of $u$ and a closed target set $\mathcal{S}$ depending both on the state and on the control $u$, we study the minimum time problem with a bound on the total variation of $u$ and $u$ constrained in a closed, convex set $U$, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function $T$. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize $T$ as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.

Citation: Monica Motta. Minimum time problem with impulsive and ordinary controls. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5781-5809. doi: 10.3934/dcds.2018252
References:
[1]

S. Aronna and F. Rampazzo, $L^1$ limit solutions for control systems, J. Differential Equations, 258 (2015), 954-979. doi: 10.1016/j.jde.2014.10.013.

[2]

M.S. AronnaM. Motta and F. Rampazzo, Infimum gaps for limit solutions, Set-Valued Var. Anal., 23 (2015), 3-22. doi: 10.1007/s11228-014-0296-1.

[3]

A. ArutyunovD. Karamzin and F. L. Pereira, On a generalization of the impulsive control concept: Controlling system jumps, Discrete Contin. Dyn. Syst., 29 (2011), 403-415. doi: 10.3934/dcds.2011.29.403.

[4]

J. P. Aubin, and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhuser Classics. Birkhuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[5]

D. Azimov and R. Bishop, New trends in astrodynamics and applications: Optimal trajectories for space guidance, Ann. New York Acad. Sci., 1065 (2005), 189-209. doi: 10.1196/annals.1370.002.

[6]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.

[7]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988), 641-656.

[8]

P. BousquetC. Mariconda and G. Treu, On the Lavrentiev phenomenon for multiple integral scalar variational problems, J. Funct. Anal., 266 (2014), 5921-5954. doi: 10.1016/j.jfa.2013.12.020.

[9]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhuser Boston, Inc., Boston, MA, 2004.

[10]

A. CatlláD. SchaefferT. WitelskiE. Monson and A. Lin, On spiking models for synaptic activity and impulsive differential equations, SIAM Rev., 50 (2008), 553-569. doi: 10.1137/060667980.

[11]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics vol. 178, Springer-Verlag, New York, 1998.

[12]

V. A. Dykhta, Impulse-trajectory extension of degenerated optimal control problems. The Lyapunov functions method and applications, IMACS Ann. Comput. Appl. Math., 8, Baltzer, Basel, (1990), 103-109.

[13]

H. Frankowska and S. Plaskacz, Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints, J. Math. Anal. Appl., 251 (2000), 818-838. doi: 10.1006/jmaa.2000.7070.

[14]

P. GajardoC.H. Ramirez and A. Rapaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856. doi: 10.1137/070695204.

[15]

M. Guerra and A. Sarychev, Fréchet generalized trajectories and minimizers for variational problems of low coercivity, J. Dyn. Control Syst., 21 (2015), 351-377. doi: 10.1007/s10883-014-9231-x.

[16]

V. I. Gurman, Optimal processes of singular control, Automat. Remote Control, 26 (1965), 783-792.

[17]

D. Y. KaramzinV. A. de OliveiraF. L. Pereira and G. N. Silva, On the properness of an impulsive control extension of dynamic optimization problems, ESAIM Control Optim. Calc. Var., 21 (2015), 857-875. doi: 10.1051/cocv/2014053.

[18]

V. F. Krotov, Global Methods in Optimal Control Theory, Monographs and Textbooks in Pure and Applied Mathematics, 195. Marcel Dekker, Inc., New York, 1996.

[19]

K. Kunisch and Z. Rao, Minimal time problem with impulsive controls, Appl. Math. Optim., 75 (2017), 75-97. doi: 10.1007/s00245-015-9324-2.

[20]

T. T. Le Thuy and A. Marigonda, Small-time local attainability for a class of control systems with state constraints, ESAIM Control Optim. Calc. Var., 23 (2017), 1003-1021. doi: 10.1051/cocv/2016022.

[21]

B. M. Miller, Optimization of dynamical systems with generalized control. (Russian) Avtomat. i Telemekh., 1989, 23-34; translation in Automat. Remote Control, 50 (1989), 733-742. doi: MR1016198.

[22]

B. M. Miller, The method of discontinuous time substitution in problems of the optimal control of impulse and discrete-continuous systems. (Russian) Avtomat. i Telemekh., 1993, 3-32; translation in Automat. Remote Control, 54 (1993), 1727-1750.

[23]

B. M. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214.

[24]

B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.

[25]

B. M. Miller and E. Ya. Rubinovich, Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations, Translation of Avtomat. i Telemekh., 2013, 56-103. Autom. Remote Control, 74 (2013), 1969-2006. doi: 10.1134/S0005117913120047.

[26]

M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.

[27]

M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996), 199-225. doi: 10.1137/S036301299325493X.

[28]

M. Motta and C. Sartori, Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Optim., 42 (2003), 789-809. doi: 10.1137/S0363012902385284.

[29]

M. Motta and C. Sartori, Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians, Nonlinear Anal., 49 (2002), Ser. A: Theory Methods, 905-927. doi: 10.1016/S0362-546X(01)00137-7.

[30]

M. Motta and C. Sartori, On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014), 957-982. doi: 10.1051/cocv/2014003.

[31]

M. Motta and C. Sartori, On $L^1$ limit solutions in impulsive control, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 1201-1218. doi: 10.3934/dcdss.2018068.

[32]

M. Motta and C. Sartori, Lack of BV bounds for impulsive control systems, J. Math. Anal. Appl., 461 (2018), 422-450. doi: 10.1016/j.jmaa.2018.01.019.

[33]

F. Rampazzo and C. Sartori, The minimum time function with unbounded controls, J. Math. Systems Estim. Control, 8 (1998), 1-34.

[34]

A. Razmadzé, Sur les solutions discontinues dans le calcul des variations, (French) Math. Ann., 94 (1925), 1-52. doi: 10.1007/BF01208643.

[35]

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.

[36]

R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, Princeton, NJ, 1997.

[37]

P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. Ⅱ. Equations of control problems with state constraints, Differential Integral Equations, 12 (1999), 275-293.

[38]

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

[39]

J. Warga, Variational problems with unbounded controls, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1966), 424-438. doi: 10.1137/0303028.

[40]

A. J. Zaslavski, Nonoccurrence of the Lavrentiev phenomenon for many optimal control problems, SIAM J. Control Optim., 45 (2006), 1116-1146. doi: 10.1137/050640370.

show all references

References:
[1]

S. Aronna and F. Rampazzo, $L^1$ limit solutions for control systems, J. Differential Equations, 258 (2015), 954-979. doi: 10.1016/j.jde.2014.10.013.

[2]

M.S. AronnaM. Motta and F. Rampazzo, Infimum gaps for limit solutions, Set-Valued Var. Anal., 23 (2015), 3-22. doi: 10.1007/s11228-014-0296-1.

[3]

A. ArutyunovD. Karamzin and F. L. Pereira, On a generalization of the impulsive control concept: Controlling system jumps, Discrete Contin. Dyn. Syst., 29 (2011), 403-415. doi: 10.3934/dcds.2011.29.403.

[4]

J. P. Aubin, and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhuser Classics. Birkhuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[5]

D. Azimov and R. Bishop, New trends in astrodynamics and applications: Optimal trajectories for space guidance, Ann. New York Acad. Sci., 1065 (2005), 189-209. doi: 10.1196/annals.1370.002.

[6]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.

[7]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988), 641-656.

[8]

P. BousquetC. Mariconda and G. Treu, On the Lavrentiev phenomenon for multiple integral scalar variational problems, J. Funct. Anal., 266 (2014), 5921-5954. doi: 10.1016/j.jfa.2013.12.020.

[9]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhuser Boston, Inc., Boston, MA, 2004.

[10]

A. CatlláD. SchaefferT. WitelskiE. Monson and A. Lin, On spiking models for synaptic activity and impulsive differential equations, SIAM Rev., 50 (2008), 553-569. doi: 10.1137/060667980.

[11]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics vol. 178, Springer-Verlag, New York, 1998.

[12]

V. A. Dykhta, Impulse-trajectory extension of degenerated optimal control problems. The Lyapunov functions method and applications, IMACS Ann. Comput. Appl. Math., 8, Baltzer, Basel, (1990), 103-109.

[13]

H. Frankowska and S. Plaskacz, Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints, J. Math. Anal. Appl., 251 (2000), 818-838. doi: 10.1006/jmaa.2000.7070.

[14]

P. GajardoC.H. Ramirez and A. Rapaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856. doi: 10.1137/070695204.

[15]

M. Guerra and A. Sarychev, Fréchet generalized trajectories and minimizers for variational problems of low coercivity, J. Dyn. Control Syst., 21 (2015), 351-377. doi: 10.1007/s10883-014-9231-x.

[16]

V. I. Gurman, Optimal processes of singular control, Automat. Remote Control, 26 (1965), 783-792.

[17]

D. Y. KaramzinV. A. de OliveiraF. L. Pereira and G. N. Silva, On the properness of an impulsive control extension of dynamic optimization problems, ESAIM Control Optim. Calc. Var., 21 (2015), 857-875. doi: 10.1051/cocv/2014053.

[18]

V. F. Krotov, Global Methods in Optimal Control Theory, Monographs and Textbooks in Pure and Applied Mathematics, 195. Marcel Dekker, Inc., New York, 1996.

[19]

K. Kunisch and Z. Rao, Minimal time problem with impulsive controls, Appl. Math. Optim., 75 (2017), 75-97. doi: 10.1007/s00245-015-9324-2.

[20]

T. T. Le Thuy and A. Marigonda, Small-time local attainability for a class of control systems with state constraints, ESAIM Control Optim. Calc. Var., 23 (2017), 1003-1021. doi: 10.1051/cocv/2016022.

[21]

B. M. Miller, Optimization of dynamical systems with generalized control. (Russian) Avtomat. i Telemekh., 1989, 23-34; translation in Automat. Remote Control, 50 (1989), 733-742. doi: MR1016198.

[22]

B. M. Miller, The method of discontinuous time substitution in problems of the optimal control of impulse and discrete-continuous systems. (Russian) Avtomat. i Telemekh., 1993, 3-32; translation in Automat. Remote Control, 54 (1993), 1727-1750.

[23]

B. M. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214.

[24]

B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.

[25]

B. M. Miller and E. Ya. Rubinovich, Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations, Translation of Avtomat. i Telemekh., 2013, 56-103. Autom. Remote Control, 74 (2013), 1969-2006. doi: 10.1134/S0005117913120047.

[26]

M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.

[27]

M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996), 199-225. doi: 10.1137/S036301299325493X.

[28]

M. Motta and C. Sartori, Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Optim., 42 (2003), 789-809. doi: 10.1137/S0363012902385284.

[29]

M. Motta and C. Sartori, Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians, Nonlinear Anal., 49 (2002), Ser. A: Theory Methods, 905-927. doi: 10.1016/S0362-546X(01)00137-7.

[30]

M. Motta and C. Sartori, On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014), 957-982. doi: 10.1051/cocv/2014003.

[31]

M. Motta and C. Sartori, On $L^1$ limit solutions in impulsive control, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 1201-1218. doi: 10.3934/dcdss.2018068.

[32]

M. Motta and C. Sartori, Lack of BV bounds for impulsive control systems, J. Math. Anal. Appl., 461 (2018), 422-450. doi: 10.1016/j.jmaa.2018.01.019.

[33]

F. Rampazzo and C. Sartori, The minimum time function with unbounded controls, J. Math. Systems Estim. Control, 8 (1998), 1-34.

[34]

A. Razmadzé, Sur les solutions discontinues dans le calcul des variations, (French) Math. Ann., 94 (1925), 1-52. doi: 10.1007/BF01208643.

[35]

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.

[36]

R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, Princeton, NJ, 1997.

[37]

P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. Ⅱ. Equations of control problems with state constraints, Differential Integral Equations, 12 (1999), 275-293.

[38]

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

[39]

J. Warga, Variational problems with unbounded controls, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1966), 424-438. doi: 10.1137/0303028.

[40]

A. J. Zaslavski, Nonoccurrence of the Lavrentiev phenomenon for many optimal control problems, SIAM J. Control Optim., 45 (2006), 1116-1146. doi: 10.1137/050640370.

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