Minimum time problem with impulsive and ordinary controls

Monica Motta

doi:10.3934/dcds.2018252

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

Monica Motta

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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988), 641-656. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	V. I. Gurman, Optimal processes of singular control, Automat. Remote Control, 26 (1965), 783-792. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	F. Rampazzo and C. Sartori, The minimum time function with unbounded controls, J. Math. Systems Estim. Control, 8 (1998), 1-34. Google Scholar
[34]	A. Razmadzé, Sur les solutions discontinues dans le calcul des variations, (French) Math. Ann., 94 (1925), 1-52. doi: 10.1007/BF01208643. Google Scholar
[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. Google Scholar
[36]	R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, Princeton, NJ, 1997. Google Scholar
[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. Google Scholar
[38]	J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1. Google Scholar
[39]	J. Warga, Variational problems with unbounded controls, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1966), 424-438. doi: 10.1137/0303028. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988), 641-656. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	V. I. Gurman, Optimal processes of singular control, Automat. Remote Control, 26 (1965), 783-792. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	F. Rampazzo and C. Sartori, The minimum time function with unbounded controls, J. Math. Systems Estim. Control, 8 (1998), 1-34. Google Scholar
[34]	A. Razmadzé, Sur les solutions discontinues dans le calcul des variations, (French) Math. Ann., 94 (1925), 1-52. doi: 10.1007/BF01208643. Google Scholar
[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. Google Scholar
[36]	R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, Princeton, NJ, 1997. Google Scholar
[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. Google Scholar
[38]	J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1. Google Scholar
[39]	J. Warga, Variational problems with unbounded controls, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1966), 424-438. doi: 10.1137/0303028. Google Scholar
[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. Google Scholar

[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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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]	Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
[17]	N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549
[18]	Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control & 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 & 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 & Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385

American Institute of Mathematical Sciences

Minimum time problem with impulsive and ordinary controls

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Minimum time problem with impulsive and ordinary controls

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors