Minimum time problem with impulsive and ordinary controls

Previous Article
$L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials
DCDS Home
This Issue
Next Article
On dispersion decay for 3D Klein-Gordon equation

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.

Keywords: Impulsive optimal control problems, existence of optimal controls, Lavrentiev phenomenon, Hamilton Jacobi equations, lower semicontinuous viscosity solutions.

Mathematics Subject Classification: Primary: 49N25, 49L25; Secondary: 34H5, 49J15.

Citation: Monica Motta. Minimum time problem with impulsive and ordinary controls. Discrete and Continuous Dynamical Systems, 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. Aronna, M. 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. Arutyunov, D. 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. Bousquet, C. 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. Catllá, D. Schaeffer, T. Witelski, E. 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. Gajardo, C.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. Karamzin, V. A. de Oliveira, F. 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. Aronna, M. 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. Arutyunov, D. 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. Bousquet, C. 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. Catllá, D. Schaeffer, T. Witelski, E. 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. Gajardo, C.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. Karamzin, V. A. de Oliveira, F. 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.

[1]	Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial and Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[2]	Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046
[3]	Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[4]	Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure and Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793
[5]	Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649
[6]	Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447
[7]	Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
[8]	Hongyong Deng, Wei Wei. Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1409-1422. doi: 10.3934/jimo.2015.11.1409
[9]	Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176
[10]	Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure and Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479
[11]	C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial and Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435
[12]	X. Xiang, Y. Peng, W. Wei. A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces. Conference Publications, 2005, 2005 (Special) : 911-919. doi: 10.3934/proc.2005.2005.911
[13]	Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137
[14]	Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311
[15]	M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223
[16]	N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549
[17]	Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
[18]	Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 245-260. doi: 10.3934/naco.2011.1.245
[19]	Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331
[20]	Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Monica Motta
on Google Scholar
Monica Motta

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Monica Motta
on Google Scholar
Monica Motta

[Back to Top]