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On dispersion decay for 3D Klein-Gordon equation
Minimum time problem with impulsive and ordinary controls
Università di Padova, Via Trieste, 63, Padova 35121, Italy |
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.
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. |
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